Be able to find the absolute maximum and minimum values on the given interval: (a) f(x)=12+4x−x2,[0,5] Solution: Abs max at x=2, Abs min at x=5 (b) f(x)=(x2−9)3,[−4,6] Solution: Abs max at x=6, Abs min at x=0 (c) f(x)=ex3+3x2−9x,[0,2] Solution: Abs max at x=2, Abs min at x=1

Answers

Answer 1

The absolute maximum and minimum values of the given functions on their respective intervals are as follows:

For [tex]f(x) = 12 + 4x - x²[/tex] on [0,5], the absolute maximum is 12 at x = 2 and the absolute minimum is -3 at x = 5.

The three given intervals that includes absolute maximum and minimum values for a function f(x) are (a) [0,5], (b) [−4,6] and (c) [0,2].

[tex]f(x)=12+4x−x2,[0,5][/tex]

[tex]f(x) = 12 + 4x - x² on [0,5][/tex]

The critical points of f(x) can be calculated as below:

f′(x) = 4 - 2x

This will be zero at x = 2. Critical points of f(x) are 2 and 5.

[tex]f(2) = 12 + 4(2) - (2)² = 12[/tex]

Abs max at [tex]x=2.f(5) = 12 + 4(5) - (5)² = -3[/tex]

Abs min at x=5.

In part (a), we calculated the critical points of the given function f(x) and calculated the values of f(x) at these points and endpoints to determine the absolute maximum and minimum values on the given interval (0,5). For the given function[tex]f(x) = (x² - 9)³[/tex] on the interval [−4,6], the critical points can be calculated as:

[tex]f′(x) = 6x(x² - 9)²[/tex]

It will be zero at x = 0 and x = ±3. The endpoints of the interval [−4,6] are -4 and 6. Now, we can calculate the values of f(x) at these critical points and endpoints to determine the absolute maximum and minimum values on the interval (−4,6). Critical points of f(x) are 0 and ±3.

[tex]f(6) = (6² - 9)³ = 729[/tex]

Abs max at[tex]x=6.f(-4) = (-4² - 9)³ = -274625[/tex]

Abs min at [tex]x=-4.f(0) = (0² - 9)³ = -729[/tex]

Abs min at x=0.

Therefore, we have calculated the critical points and values of f(x) at these points and endpoints for the given function [tex]f(x) = ex³ + 3x² − 9x[/tex] on the interval [0,2] to determine the absolute maximum and minimum values on the given interval. The maximum and minimum values are as follows: Critical point of f(x) is 1.

[tex]f(2) = e(2)³ + 3(2)² − 9(2) = e⁸ - 18[/tex]

Abs max at x=2.[tex]f(1) = e(1)³ + 3(1)² − 9(1) = e - 6[/tex]

Abs min at x=1.

Therefore, the absolute maximum and minimum values of the given functions on their respective intervals are as follows:

For [tex]f(1) = e(1)³ + 3(1)²[/tex]on [0,5], the absolute maximum is 12 at x = 2 and the absolute minimum is -3 at x = 5.

For f(x) on [−4,6], the absolute maximum is 729 at x = 6 and the absolute minimum is -274625 at x = -4.

For[tex]f(x) = ex³ + 3x² − 9x[/tex] on [0,2], the absolute maximum is e⁸ - 18 at x = 2 and the absolute minimum is e - 6 at x = 1.

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Related Questions

5:3 6528 votes how many were yes

Answers

There were 2448 votes against the issue.

The ratio of the votes for a particular issue in a poll was 5:3. There were 6528 votes in total. Find out the number of votes in favor of the issue.

Follow the below steps to find out the number of votes in favor of the issue:

Step 1: Add the values of the ratio: 5 + 3 = 8

Step 2: Divide the total votes by the value of the ratio: 6528 / 8 = 816

Step 3: Multiply the numerator of the ratio by the quotient obtained in step 2 to find the number of votes in favor of the issue: 5 × 816 = 4080Therefore, out of 6528 votes in the poll, 4080 were in favor of the issue.

You can also calculate the number of votes against the issue by multiplying the numerator of the ratio by the quotient obtained in step 2: 3 × 816 = 2448.

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A sector of a circle has a central angle of 120 degrees. Find
the area of the sector if the radius of the circle is 17 cm.
-answer in cm^2

Answers

Given that the central angle of a sector of a circle is 120° and the radius of the circle is 17 cm.

Area of a sector of a circle is given as: Area of sector.

= (θ/360°)πr²

θ =is the central angle and r being the radius of the circle.

Substitute the given values of θ and r in the above formula, we get:

Area of sector

= (120°/360°)π(17) ²

= (1/3)π(289)

= 289π/3 cm²

=96.02 cm²

Therefore, the area of the sector is 96.02 cm² (rounded off to two decimal places).

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For the demand function d(x) and demand level x, find the consumers' surplus. d(x)=900− 2
1
​ x,x=

Answers

Therefore, the consumer's surplus is given by the function [tex]CS(x) = 900x - x^2.[/tex]

To find the consumer's surplus, we need to calculate the integral of the demand function over the range from 0 to x.

The given demand function is:

d(x) = 900 - (2/1) * x

To find the consumer's surplus, we integrate the demand function from 0 to x:

CS = ∫[0,x] d(x) dx

CS = ∫[0,x] (900 - (2/1) * x) dx

[tex]CS = [900x - (2/2) * x^2][/tex] evaluated from 0 to x

[tex]CS = (900x - x^2) - (900(0) - (0)^2)[/tex]

[tex]CS = 900x - x^2[/tex]

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Evaluate the integral \( \int_{1}^{2} \frac{4 x^{2}-3 x+4}{x} d x \) a. \( 9+4 \ln 3 \) b. \( 3+4 i n 2 \) c. \( 9+2 \ln 4 \) d. \( 3+2 \ln 4 \)

Answers

The value of the integral [tex]\( \int_{1}^{2} \frac{4 x^{2}-3 x+4}{x} d x \)[/tex] is 9 + 4ln2.

the correct answer is (a) 9+4ln2.

Here, we have,

To evaluate the integral [tex]\( \int_{1}^{2} \frac{4 x^{2}-3 x+4}{x} d x \)[/tex]

we can use the properties of logarithms.

First, we rewrite the integrand as:

4x - 3 + 4/x

Now, we can integrate each term separately:

∫₁² 4x dx - ∫₁² 3 dx + ∫₁²4/x dx

Integrating each term:

2x² - 3x + 4 ln|x|  [from 1 to 2]

Evaluating each term:

we get,

8 - 6 + 4 ln2 - 2 - 3 + 0

= 9 + 4 ln2

Therefore, the correct answer is (a) 9+4ln2.

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complete question:

Evaluate the integral [tex]\( \int_{1}^{2} \frac{4 x^{2}-3 x+4}{x} d x \)[/tex]

[tex]a. \( 9+4 \ln 3 \) \\b. \( 3+4 i n 2 \) \\c. \( 9+2 \ln 4 \) \\d. \( 3+2 \ln 4 \)[/tex]

21
The curved parts of the figure are arcs centered at points A and C. What is the approximate length of boundary ABCD? Use the value = 3.14, and
round the answer to one decimal place.
A
5
120*
30°

Answers

Answer: 21.9

Step-by-step explanation: To find the length of boundary ABCD, we add the lengths of each line segment and the two arcs. AB is 5, BC is (120/360) * 2 * pi * 5 = 10pi/3, CD is 5, and DA is (30/360) * 2 * pi * 5 = pi/3. Adding these lengths, we get (20pi + 15)/3, which is approximately 21.9 when rounded to one decimal place.

- Lizzy ˚ʚ♡ɞ˚

Find f'(x) when f(x) = e^x^3 + xin(x^2)

Answers

Given function is,[tex]f(x) = e^(x^3) + x * in(x^2)[/tex]Let's find the derivative of the given function. We need to use the product rule of differentiation in order to find the derivative of the function, as it is a product of two functions.

Product Rule:

[tex]f(x) = u(x) * v(x)\\f'(x) = u'(x) * v(x) + u(x) * v'(x)\\Let's take u(x) = e^(x^3) and \\v(x) = x * in(x^2)\\u'(x) = d/dx(e^(x^3)) \\= 3x^2 * e^(x^3)[/tex]

The derivative of ln(x^2) is given as :

[tex][x * 2x * e^(x^3)] + [e^(x^3)] * [ln(x^2) + 1/x]f'(x) = [3x^3 * ln(x^2) * e^(x^3)] + [2x^2 * e^(x^3)] + [e^(x^3)] * [ln(x^2) + 1/x]\\Therefore, f'(x) = [3x^3 * ln(x^2) * e^(x^3)] + [2x^2 * e^(x^3)] + [e^(x^3)] * [ln(x^2) + 1/x][/tex]

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A and B are independent events. Use the following probabilities to answer the question. Round to 4 decimal places. P(A) 0.57, P(A and B) = 0.34, find P(B)

Answers

The probability of event B occurring is 0.5965. In probability theory, two events are independent if the occurrence of one event does not affect the probability of the other event occurring.

If A and B are two independent events, then the probability of both events occurring at the same time is given by the product of their individual probabilities.

In this case, we know that P(A) = 0.57 and P(A and B) = 0.34. We need to find P(B).

We can use the formula for the probability of the intersection of two events, P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A has occurred. Since A and B are independent, P(B|A) = P(B). Substituting the given values, we get:

0.34 = 0.57 × P(B)
P(B) = 0.34 / 0.57
P(B) = 0.5965 (rounded to 4 decimal places)

Therefore, the probability of event B occurring is 0.5965.

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A computer is programmed to take the sum of 400 draws made at random with replacement from the box 0 2 4 6.
a. What are the lowest and highest possible values for the sum of the 400 draws?
Highest _____ Lowest _____
b. What is the expected value and the standard error for the sum of the 400 draws?
Expected value for the sum _____ SE for the sum _____
c. What is the expected value and standard error for the average of the 400 draws?
Expected value for the average _____ SE for the average _____

Answers

a. What are the lowest and highest possible values for the sum of the 400 draws?Highest value = 400 × 6 = 2400Lowest value = 400 × 0 = 0b. What is the expected value and the standard error for the sum of the 400 draws?The mean and variance of a single draw are given by:

Mean = (0 + 2 + 4 + 6) / 4 = 3Var = [(0 − 3)² + (2 − 3)² + (4 − 3)² + (6 − 3)²] / 4 = 3.

The expected value of the sum is:

Mean of sum = 400 × 3 = 1200.

The variance of the sum is:Var of sum = 400 × 3 = 1200The standard error for the sum is:

SE for sum = sqrt(Var of sum) = sqrt(1200) ≈ 34.6c.

What is the expected value and standard error for the average of the 400 draws?

The expected value of the average is:

Mean of average = Mean of sum / number of draws = 1200 / 400 = 3.

The variance of the average is:

Var of average = Var of sum / (number of draws)²= 1200 / (400)² = 0.075.

The standard error for the average is: SE for average = sqrt(Var of average) = sqrt(0.075) ≈ 0.27.

The highest possible value for the sum of the 400 draws is 2400, while the lowest possible value is 0. The expected value and the standard error for the sum of the 400 draws are 1200 and 34.6 respectively. The expected value and standard error for the average of the 400 draws are 3 and 0.27 respectively.

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Homework for Section \( 8.2 \) Score: \( 27 / 32 \quad 6 / 8 \) answered Assume that a sample is used to estimate a population proportion p. Find the \( 80 \% \) confidence interval for a sample of si

Answers

The critical value for an 80% confidence level is approximately 1.282.

To find an 80% confidence interval for a sample proportion, we can use the formula:

Confidence interval = sample proportion ± (z* * standard error)

where the sample proportion is denoted as p-hat, z* is the critical value corresponding to the desired confidence level, and the standard error is calculated using the formula:

Standard error = sqrt((p-hat * (1 - p-hat)) / n)

In this case, we are given the sample size (n), the sample proportion (p-hat), and the desired confidence level (80%).

We need to find the critical value, which corresponds to the remaining percentage (100% - 80% = 20%) divided by 2 (to split the remaining percentage equally in the two tails of the distribution).

Using a standard normal distribution table or a statistical calculator, we can find that the critical value for an 80% confidence level is approximately 1.282 (rounded to three decimal places).

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Ashley had 4/ 5 of a spool of yarn. She used 2/5 of it for her project. What fraction of the spool was used for her project? Write your answer in simplest form

Answers

Ashley used 8/25 of the spool for her project.

To determine the fraction of the spool that Ashley used for her project, we need to multiply the fraction of the spool she had (4/5) by the fraction she used (2/5):

(4/5) * (2/5) = 8/25

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System Schematic A regenerative gas turbine with intercooling and reheat operates at steady state. Air enters the compressor at 100 kPa, 300 K with a mass flow rate of 5.807 kg/sec. The pressure ratio across the two-stage compressor is 10. The intercooler and reheater each operate at 300 kPa. At the inlets to the turbine stages, the temperature 1400 K. The temperature at the inlet to the second compressor is 300 K. The isentropic efficiency of each compressor stage and turbine stage is 80%. The regenerator effectiveness is 80%. Given: P1 = P9 = P10 = 100 KPa P2 P3 300 kPa P4 P5 P6= 1000 kPa T1 T3 = 300 K nst = 80% nsc = 80% Ts 1400 K T6 P7 P8 300 kPa m = 5.807 kg/sec Engineering Model: 1- CV-SSSF 2 - qt=qc = 0 3 - Air is ideal gas. 4 - AEk,p=0 System Schematic: Figure E9.11

Answers

The notation used is P1 = P9 = P10 = 100 KPa P2 P3 300 kPa P4 P5 P6= 1000 kPa T1 T3 = 300 K nst = 80% nsc = 80% Ts 1400 K T6 P7 P8 300 kPa m = 5.807 kg/sec.

Regenerative Gas Turbine is a machine used for the generation of electricity, directly through combustion process (burning of natural gas, coal, and oil), and indirectly through Steam Generation (by using waste heat generated in the turbine).It comprises of a compressor, combustor, and a turbine.

The input air to the compressor is compressed and sent into the combustor. Fuel and compressed air are burnt here, and the resulting hot gases then expand through the turbine, which drives the generator and produces electricity.The given Regenerative Gas Turbine with intercooling and reheating process at steady-state.

Air enters the compressor at 100 kPa, 300 K with a mass flow rate of 5.807 kg/sec. The pressure ratio across the two-stage compressor is 10. The intercooler and reheater each operate at 300 kPa.The temperature at the inlet to the second compressor is 300 K. The isentropic efficiency of each compressor stage and turbine stage is 80%. The regenerator effectiveness is 80%.

The notation used is P1 = P9 = P10 = 100 KPa P2 P3 300 kPa P4 P5 P6= 1000 kPa T1 T3 = 300 K nst = 80% nsc = 80% Ts 1400 K T6 P7 P8 300 kPa m = 5.807 kg/sec, and the Engineering Model used are CV-SSSF, qt=qc=0, Air is ideal gas, and AEk,p=0.

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4. Let f(x) = 2x³ – 9x² − 38x + 21. - (a) List all possible rational roots of f(x). (b) Factor f(x) completely. (c) Sketch a rough graph of f(x). Make sure the x-intercepts are labeled.

Answers

The factored form of f(x) is (2x - 1)(x + 3)(x - 7).

To find the possible rational roots of the polynomial f(x) = 2x³ - 9x² - 38x + 21, we can use the Rational Root Theorem. According to the theorem, the possible rational roots are all the divisors of the constant term (21 in this case) divided by the divisors of the leading coefficient (2 in this case). Let's find the possible rational roots:

(a) Possible rational roots of f(x):

±1, ±3, ±7, ±21

To factor f(x) completely, we can use the possible rational roots obtained in part (a) and perform synthetic division or long division to find the factors. However, in this case, the polynomial is already given, so we can directly factor it:

(b) Factored form of f(x):

f(x) = (2x - 1)(x + 3)(x - 7)

(c) Rough graph of f(x):

To sketch a rough graph of f(x), we can plot the x-intercepts corresponding to the roots we found earlier: x = 1/2, x = -3, and x = 7. Additionally, we can analyze the leading coefficient and the degree of the polynomial to determine the behavior of the graph. Since the leading coefficient is positive and the degree is odd (3 in this case), the graph will start from the bottom left quadrant and go up towards the top right quadrant.

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A random sample (i.i.d) of size 5 is drawn from the pdf fy (y) = 2y. 0 ≤ y ≤ 1. Let Y' be the i-th order statistic. Compute P(Y> 0.6). Compute P(Y<0.6). Compute P(Y < 0.6 < Y'). The events (Y> 0.6) and (Y' < 0.6) are events, therefore the probability of the union is the

Answers

The answer is 1.1474.

A random sample (i.i.d) of size 5 is drawn from the pdf fy (y) = 2y. 0 ≤ y ≤ 1.

Let Y' be the i-th order statistic.

We need to calculate:[tex]P(Y> 0.6), P(Y<0.6), P(Y < 0.6 < Y').P(Y> 0.6)If Y > 0.6,[/tex]

then Fy(y) = P(Y < y) = 2∫y0tdt = y2for 0 ≤ y ≤ 1.

Thus,[tex]P(Y > 0.6) = 1 - P(Y < 0.6) = 1 - Fy(0.6) = 1 - (0.6)2 = 0.64P(Y<0.6)[/tex]

If Y < 0.6, then Fy(y) = P(Y < y) = 2∫y0tdt = y2 for 0 ≤ y ≤ 1.

Thus,[tex]P(Y < 0.6) = Fy(0.6) = (0.6)2 = 0.36P(Y < 0.6 < Y')P(Y < 0.6 < Y') = P(Y' > 0.6) - P(Y' < 0.6) = [1 - P(Y' < 0.6)] - P(Y' < 0.6) = 1 - 2P(Y' < 0.6)[/tex]

Here, Y' is the ith order statistic, so for i = 5, we have the exponential distribution with f(y) = 10e-10y and F(y) = 1 - e-10y

So, [tex]P(Y' < 0.6) = F(0.6) = 1 - e-10(0.6) = 0.5474[/tex]

Therefore[tex],P(Y < 0.6 < Y') = 1 - 2(0.5474) = -0.0948[/tex]

The events (Y> 0.6) and (Y' < 0.6) are events, therefore the probability of the union is the sum of their probabilities, minus the probability of their intersection.

P(Y > 0.6 ∪ Y' < 0.6) = P(Y > 0.6) + P(Y' < 0.6) - P(Y > 0.6 ∩ Y' < 0.6)

But, P(Y > 0.6 ∩ Y' < 0.6) = P(Y' < Y < 0.6) = ∫0.60∫yy2dydx = ∫00.6y2dx∫y0.6dy = 1/125

Thus,P(Y > 0.6 ∪ Y' < 0.6) = 0.64 + 0.5474 - 1/125 = 1.1474

The answer is 1.1474.

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The Ohio Department of Education maintains records of average number of years of teaching experience for each public school in the state. During the 2012-2013 school year, it was reported that the average number of years of teaching experience at Ohio high schools was 14.3 years. Suppose that an intern working in educational policy research wants to determine whether the average number of years of teaching experience of teachers in Ohio high schools changed between the 2012-2013 and 2013-2014 school years. The intern selected a random sample of 13 high schools, and average number of years of teaching experience for the 2013 2014 school year at each of these 13 schools is recorded below. Prior years' data suggest that mean teaching experience at Ohio public high schools is normally distributed. 12,16,7,11,10,15,20,12,11,15,12,15,13 If you wish, you may download the data in your preferred format. CrunchIt! CSV Excel JMP Mac Text Minitab14-18 Minitab18+ PC Text R SPSS TI Calc Use a two-tailed one-sample t-test to determine whether average number of years of teaching experience at Ohio high schools during the 2013-2014 school year was different from 14.3 years. Have the requirements for a one-sample t-test been met? If they have not been met, leave the remaining questions blank. a. Yes, the intern selected a random sample from a normally distributed population, and his sample contains no outliers. b. Yes, the intern selected a random sample that is normally distributed and contains no outliers c. No, the intern selected a random sample from a normally distributed population, but his sample is too small.

Answers

The requirements for a one-sample t-test have not been met because the sample size is too small to assume a normal distribution and to detect outliers effectively.

The intern selected a random sample of 13 high schools, which is not large enough to assume that the sampling distribution of the mean is approximately normal. Additionally, the sample size may not be sufficient to identify potential outliers that could affect the results.

Therefore, the requirements for a one-sample t-test have not been satisfied.

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Orlando skipped rope 135 times in 45 seconds. Write this rate as a unit rate.

nd

Answers

Orlando skipped rope at a rate of 3 skips per second. This means that, on average, he completed three skips every second during the 45-second time frame.

To calculate the unit rate of Orlando's skipping rope, we divide the total number of times he skipped (135) by the total time it took (45 seconds). The unit rate is a ratio that compares two different units in a 1:1 relationship.

In this case, we want to find the number of times Orlando skips per second.

To convert the given rate into a unit rate, we divide the total number of skips by the total time:

Unit Rate = Total Skips / Total Time

Unit Rate = 135 skips / 45 seconds

Simplifying this ratio, we get:

Unit Rate = 3 skips / 1 second

Unit rates are useful for comparing quantities and making calculations easier, as they provide a standardized measure for comparison.

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A survey found that 13% of companies are downsizing due to the effect of the Covid-19 pandemic. A sample of five companies is selected at random. i. Find the average and standard deviation of companies that are downsizing. (3 marks) ii. Is it likely that THREE (3) companies are downsizing? Justify your answer. (4 marks) b) It is believed that sufferers of a cold virus experience symptoms for seven days. However, the total number of days is a normally distributed random variable whose mean is 7.5 days and the standard deviation is 1.2 days. i. What is the probability of a cold sufferer experiencing symptomis for at least FOUR (4) days? (4 marks) ii. What is the probability of a cold sufferer experiencing symptoms between SEVEN (7) and TEN (10) days? (5 marks) iii. What is the minimum number of days in which 67% of cold sufferers experience

Answers

The standard deviation of the number of companies that are downsizing is approximately 0.817.

The probability of exactly three companies downsizing is approximately 0.275.

To find the average and standard deviation of companies that are downsizing, we use the properties of a binomial distribution. Let's denote the number of companies that are downsizing as a random variable X.

that 13% of companies are downsizing, we have p = 0.13 as the probability of success for each trial (company being downsized) and q = 1 - p = 0.87 as the probability of failure.

The average or expected value of X is given by E(X) = np, where n is the number of trials (sample size). In this case, n = 5.

E(X) = 5 * 0.13 = 0.65

So, the average number of companies that are downsizing is 0.65.

The standard deviation of X is given by σ = √(npq).

σ = √(5 * 0.13 * 0.87) ≈ 0.817

So, the standard deviation of the number of companies that are downsizing is approximately 0.817.

ii. To determine whether it is likely that three companies are downsizing, we can calculate the probability of exactly three successes in five trials using the binomial distribution.

P(X = 3) = (5 choose 3) * 0.13^3 * 0.87^2

P(X = 3) = (5! / (3! * 2!)) * 0.13^3 * 0.87^2

P(X = 3) = 10 * 0.13^3 * 0.87^2 ≈ 0.275

The probability of exactly three companies downsizing is approximately 0.275.

Therefore, it is likely that three companies are downsizing, considering the probability is not extremely low or extremely high.

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Consider the following relation. Step 3 of 3: Determine the implied domain of the function found in the first step. Express your answer in interval notation. Answer f(x) = -3x² 2 ((-3)==- - 3x² - 2x

Answers

The implied domain of the function is (-∞, ∞).

To determine the implied domain of the function f(x) = -3x² - 2x, we need to find the set of all possible input values for x that would yield valid output values.

The function is a polynomial, and there are no restrictions on the domain of a polynomial function. Therefore, the implied domain of f(x) = -3x² - 2x is all real numbers, (-∞, ∞), expressed in interval notation.

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Compute the partial derivative: f(x, y) = sin(x6 - 6y) fy(0, π) = ⠀ gs The plane y 1 intersects the surface z = x² + 3xy - y¹ in a certain curve. Find the slope of the tangent line of this curve at the point P = (1, 1, 3). m =

Answers

Given: The function `f(x, y) = sin(x6 - 6y)`Find the partial derivative: `fy(0, π)`Now, let's begin with the given function: `f(x, y) = sin(x^6 - 6y)`

To find `fy(0, π)`, we need to find the partial derivative of the function `f` w.r.t `y`.So, `fy(x, y) = -6 cos(x^6 - 6y)`Hence, `fy(0, π) = -6 cos(0 - 6π) = -6 cos(6π) = -6`

Therefore, the value of `fy(0, π) = -6` The equation of the plane is given as y = 1, and the surface is given as z = x² + 3xy - y². Therefore, we can say that the curve in which the plane intersects the surface can be expressed as:`z = x² + 3x(1) - 1²` => `z = x² + 3x - 1`

Now, we need to find the slope of the tangent line of this curve at the point P = (1, 1, 3).For this, we need to find the first partial derivatives of the function w.r.t `x` and `y`.`∂z/∂x = 2x + 3``∂z/∂y = 0`At point P = (1, 1, 3), we get:`∂z/∂x` at `(1, 1, 3) = 2(1) + 3 = 5` Now, we need to find the direction of the tangent line at point P, and for this, we need to take the gradient of the function w.r.t `x` and `y`.grad(f) = (2x + 3)i + 0j + (-2y)kNow, putting the values of x = 1 and y = 1, we get:grad(f) at (1, 1, 3) = (5i - 2k)We know that the slope of the tangent line is equal to the magnitude of the gradient vector.

Therefore, the slope of the tangent line at point P = (1, 1, 3) is given by:m = |grad(f) at (1, 1, 3)| = √(5² + 0² + (-2)²) = √29. Hence, the slope of the tangent line of the curve at point P = (1, 1, 3) is `m = √29`

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A tank with volume 2 m³ is filled with oil whose specific gravity is 0.85. calculate the specific weight. A liquid with specific gravity 0.85 is filled a tank and its mass is 1700000 g. calculate the specific weight, specific volume and volume. Determine the density, specific gravity and mass of gas in a room whose dimension 4m x 5m x 6m at 100 kpa and 25 °c. R= 0.287 (kpa. m³/kg. K)

Answers

The specific weight of the oil in the tank can be calculated by multiplying the specific gravity of the oil by the acceleration due to gravity. In this case, the specific gravity is given as 0.85. The specific weight is equal to 0.85 times the acceleration due to gravity, which is approximately 9.8 m/s². Therefore, the specific weight of the oil is 8.33 kN/m³.

To calculate the specific volume of the oil, we need to divide the volume of the tank by the mass of the oil. The mass of the oil can be calculated by converting the given mass of 1700000 g to kilograms (1700 kg). The specific volume is equal to the volume of the tank divided by the mass of the oil, which is 2 m³ divided by 1700 kg. Therefore, the specific volume of the oil is approximately 0.0012 m³/kg.

The volume of the oil can be calculated by multiplying the specific volume by the mass of the oil. In this case, the specific volume is 0.0012 m³/kg and the mass is 1700 kg. Therefore, the volume of the oil is 0.0012 m³/kg multiplied by 1700 kg, which is approximately 2.04 m³.

To determine the density of the gas in the room, we can use the ideal gas law. The ideal gas law states that the density of a gas is equal to the product of its pressure, molar mass, and temperature divided by the gas constant. In this case, the pressure is given as 100 kPa, the molar mass is unknown, and the temperature is 25 °C. We can convert the temperature to Kelvin by adding 273.15, which gives us 298.15 K. The gas constant is given as 0.287 kPa·m³/kg·K.

We can rearrange the ideal gas law equation to solve for the molar mass of the gas. The molar mass is equal to the density multiplied by the gas constant, divided by the product of the pressure and temperature. Substituting the given values, we have molar mass = (density * 0.287) / (100 * 298.15). Therefore, the molar mass of the gas in the room can be calculated using this equation.

The specific gravity of a gas is defined as the ratio of its density to the density of a reference substance, usually air at a specific temperature and pressure. The specific gravity can be calculated by dividing the density of the gas by the density of the reference substance. Therefore, the specific gravity of the gas in the room can be calculated using the density of the gas and the density of air.

The mass of the gas in the room can be calculated by multiplying the density of the gas by the volume of the room. In this case, the volume of the room is given as 4m x 5m x 6m, which is 120 m³. Therefore, the mass of the gas can be calculated by multiplying the density of the gas by 120 m³.

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The Derivative F′(X) Is Graphed In The Figure. Fill In The Table Of Values For F(X) Given That F(0)=−10.

Answers

The table of values for f(x) is:

x | f(x)

0 | -10

1 | -10

2 | -6

3 | 10

Since the derivative function f'(x) is given, we can find the corresponding original function f(x) by integrating f'(x) with respect to x.

To fill in the table of values for f(x), we start with the initial condition f(0) = -10.

x | f'(x)

0 | -10

To find f(x) for other values of x, we integrate f'(x) term by term:

∫ f'(x) dx = ∫ 3x^2 - 6x + 4 dx

Integrating each term separately:

∫ 3x^2 dx = x^3 + C1

∫ -6x dx = -3x^2 + C2

∫ 4 dx = 4x + C3

Adding the constants of integration C1, C2, and C3, we get:

f(x) = x^3 - 3x^2 + 4x + C

To determine the value of the constant C, we use the initial condition f(0) = -10:

-10 = (0)^3 - 3(0)^2 + 4(0) + C

-10 = 0 + 0 + 0 + C

C = -10

Therefore, the function f(x) is:

f(x) = x^3 - 3x^2 + 4x - 10

Now we can fill in the table of values for f(x) using the derived function:

x | f(x)

0 | -10

1 | -10

2 | -6

3 | 10

The table of values for f(x) is:

x | f(x)

0 | -10

1 | -10

2 | -6

3 | 10

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represents simulated birthdates of 15 individuals 2000 times. Each birthdate was assigned a number from 1 (January 1 ) to 365 (December 31 ). Complete parts (a) through (c) below. Click the icon to view the links to the data file. Choose data set 1 g. (a) What is the birthdate of the randomly selected individual in row 2 , column 2?

Answers

The data table represents simulated birthdates of 15 individuals 2000 times. Each birthdate was assigned a number from 1 (January 1 ) to 365 (December 31 ). To find the birthdate of the randomly selected individual in row 2, column 2, we need to follow the given steps:

Step 1: Open the data file and select data set 1g.Step 2: Find the row 2 and column 2.Step 3: Find the corresponding value in the cell where row 2 and column 2 intersect. The value in this cell represents the birthdate of the randomly selected individual in row 2, column 2.

The value in this cell is 297, which represents October 24.To represent the birthdate of each individual 2000 times, there are 2000 columns. Thus, each row represents the birthdates of a particular individual.

Since we need to find the birthdate of the randomly selected individual in row 2, column 2, we need to look for the value in the second row of the second column of the given data set.

Therefore, the birthdate of the randomly selected individual in row 2, column 2 is October 24.

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Find the area of the surface generated when the given curve is revolved about the y-axis. y = 2, for 4 ≤ x ≤ 6 4r(103/2-53/2) (103/2-53/2) (73/2-63/2) 87 (103/2-53/2)

Answers

The area of the surface generated is -50π square units.

To find the area of the surface generated when the curve y = [tex]x^2[/tex]/4 is revolved about the y-axis, we can use the formula for the surface area of revolution:

A = 2π∫[a,b] x * [tex]\sqrt{[/tex](1 + [tex](dy/dx)^2[/tex]) dx

In this case, we need to find dy/dx to substitute it into the formula.

Given the curve y = [tex]x^2[/tex]/4, we can differentiate both sides with respect to x to find dy/dx:

dy/dx = (1/4) * 2x

= x/2

Now we can substitute this into the surface area formula and integrate over the interval [4, 6]:

A = 2π∫[4,6] x * [tex]\sqrt{[/tex](1 + [tex](x/2)^2[/tex]) dx

To evaluate this integral, we can make the substitution u = 1 + [tex](x/2)^2[/tex], which gives us du = (1/2) * x dx. Rearranging this, we have x dx = 2 du.

Substituting the new variables and limits of integration, the integral becomes:

A = 2π∫[u(4),u(6)] (2u - 2) du

Simplifying further:

A = 4π∫[u(4),u(6)] (u - 1) du

Now we integrate with respect to u:

A = 4π[([tex]u^2[/tex]/2) - u] evaluated from u = u(4) to u = u(6)

To find the values of u(4) and u(6), substitute the corresponding x-values into the equation u = 1 + [tex](x/2)^2[/tex]:

u(4) = 1 + [tex](4/2)^2[/tex] = 5

u(6) = 1 + [tex](6/2)^2[/tex] = 10

Substituting these values back into the surface area equation:

A = 4π[([tex]5^2[/tex]/2) - 5 - ([tex]10^2[/tex]/2) + 10]

= 4π[(25/2) - 5 - (100/2) + 10]

= 4π[(25/2) - (10) - (50) + 10]

= 4π[-25/2]

= -50π

Therefore, the area of the surface generated when the curve y = [tex]x^2[/tex]/4 is revolved about the y-axis is -50π square units. Note that the negative sign indicates that the surface is oriented in the opposite direction.

Correct Question :

Find the area of the surface generated when the given curve is revolved about the y-axis.

y = [tex]x^2[/tex]/4, for 4 ≤ x ≤ 6

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Multiply and simplify [(sinθ+cosθ)(sinθ+cosθ)−1]/(sinθcosθ)

Answers

The answer of expression  is 2/tanθ.

Given expression is:(sinθ + cosθ)(sinθ + cosθ) − 1 / (sinθcosθ)

Let's simplify it first:(sin² θ + 2 sin θ cos θ + cos² θ - 1) / (sinθcosθ)

Now, (sin² θ + cos² θ = 1) and (2 sin θ cos θ = sin 2θ)

Putting these values in the simplified expression, we get:(1 + sin 2θ - 1) / (sinθcosθ)sin 2θ / (sinθcosθ)

Multiply and simplify:2 / tanθ

The answer is 2/tanθ.

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A ball is thrown upward at an angle of 60° to the ground. If the ball lands 120 m away, what was the initial speed of the ball? (12 points) (You may use g in your computation, no need to use g = 9.8 m/s².)

Answers

The initial velocity of the ball is approximately 145.2 m/s.To determine the initial velocity of a ball thrown upward at an angle of 60° to the ground, which lands 120 m away, use the following steps

The given values are:θ = 60°s = 120 mWe know that the horizontal velocity (vx) is given as:vx = s / t Since the ball lands at the same height it was thrown from, the time of flight of the ball is given as:t = 2u sin θ / g (time of flight equation)where g = 9.8 m/s² (acceleration due to gravity)

The vertical velocity (vy) can be determined using the following formula: v = u sin θ - gt (velocity equation)

Finally, the initial velocity of the ball (u) can be determined using the Pythagorean theorem, which states that the hypotenuse of a right triangle (in this case, the initial velocity) is given by the square root of the sum of the squares of the other two sides (in this case, vx and vy).

This can be expressed as:u = sqrt(vx² + vy²)

Therefore, we have:vx = s / t = s / [2u sin θ / g]= g * s / [2u sin θ]vy = u sin θ - gt = u sin θ - g(2u sin θ / g)= u sin θ - 2u sin θ= - u sin θu = sqrt(vx² + vy²) = sqrt[(g * s / 2u sin θ)² + (- u sin θ)²]= sqrt[g²s² / (4u² sin²θ) + u² sin²θ]

Multiplying through by 4u² sin²θ gives: 4u⁴ sin⁴θ + 4u² g² s² sin²θ = 16u⁴ sin⁴θ

Substituting w = u² and solving for w:w² - 4g² s² sin²θ w = 0w = 4g² s² sin²θ (since w cannot be negative)

Therefore, we have:w = u² = 4g² s² sin²θu = sqrt(4g² s² sin²θ)= 2g s sin θ= 2(9.8 m/s²)(120 m) sin 60°≈ 145.2 m/s

Therefore, the initial velocity of the ball is approximately 145.2 m/s.  

The initial velocity of the ball is approximately 145.2 m/s.

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The claim is that for 7 AM body temperatures of females ​, the mean is less than 98.6 degrees Upper F . The sample size is nequals 36 and the test statistic is tequals negative 4.059. Use technology to find the​ P-value. Based on the​ result, what is the final​ conclusion? Use a significance level of 0.01 .
state the null and alternative hypotheses
H0
H1
The test statistic is
enter your response here
.
​(Round to two decimal places as​ needed.)
Part 3
The​ P-value is
enter your response here

Answers

The null and alternative hypotheses are given as below:

H0: μ = 98.6H1: μ < 98.6The claim is that for 7 AM body temperatures of females,  

The mean is less than 98.6 degrees Upper F.

The sample size is n= 36 and the test statistic is t= -4.059

. We need to use technology to find the P-value.  

Based on the result, what is the final conclusion using a significance level of 0.01.

The P-value is: The P-value for the left-tailed test is the probability that the t-statistic is less than -4.059 when the degree of freedom is 35.

Using a calculator, the P-value is 0.0002.

This means that there is a 0.02% chance of observing a t-statistic less than -4.059 due to chance alone.

The P-value is less than 0.01 (significance level),

We reject the null hypothesis and conclude that there is sufficient evidence to support the claim that for 7 AM body temperatures of females,

The mean is less than 98.6 degrees Upper F.

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When mudcake builds up on the borehole wall, this can prevent further invasion of the formation Select one: True False

Answers

Mudcake buildup on the borehole wall can indeed prevent further invasion of the formation is a true statement.

When Mudcake builds up on the borehole wall, it can create a barrier that hinders or prevents further invasion of the formation.

Mudcake is a layer of mud solids and other additives that forms on the borehole wall during the drilling process.

It serves as a filter cake, helping to control fluid loss and stabilize the wellbore.

However, if the Mudcake becomes too thick or dense,

it can effectively block the pores in the formation and restrict the flow of fluids into the wellbore.

This can prevent further invasion of the formation by drilling fluids or other fluids used in well operations.

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Find an equation for the conic section with the given properties. The ellipse with vertices V₁ (-1,-4) and V₂(-1,6) and foci F₁ (-1,-3) and F₂ (-1,5) 40. The parabola with focus F(1,3) and directrix x=3

Answers

The equation for the parabola is: 4(1)(y - 3) = (x - 2)^2

The equation for the ellipse with the given properties is:

((x + 1)^2 / a^2) + ((y - 1)^2 / b^2) = 1

where a represents the semi-major axis and b represents the semi-minor axis of the ellipse.

To find the values of a and b, we can use the distances between the vertices and foci. The distance between the vertices is 10, and the distance between the foci is 2. This relationship holds for ellipses, where the sum of the distances from any point on the ellipse to the foci is constant.

Using these distances, we can determine that a = 5 and b = √21.

Therefore, the equation for the ellipse is:

((x + 1)^2 / 25) + ((y - 1)^2 / 21) = 1

The equation for the parabola with the given properties is:

4p(y - k) = (x - h)^2

where p represents the distance from the vertex to the focus (which is also the distance from the vertex to the directrix), and (h, k) represents the coordinates of the vertex.

From the given information, the focus is F(1,3) and the directrix is x=3. The vertex is the midpoint between the focus and directrix, so the vertex is V(2,3).

The distance from the vertex to the focus (or directrix) is the value of p. In this case, p = 1.

Simplifying, we have:

4(y - 3) = (x - 2)^2

This is the equation for the parabola.

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The position vector for a particle moving on a helix is c(t) = (5 cos(1), 5 sin(t), 12). Find the speed s(to) of the particle at time to = 7. (Express numbers in exact form. Use symbolic notation and fractions where needed.) s(to) = 25+ 196 2 Find parametrization for the tangent line at time to = 7л. Use the equation of the tangent line such that the point of tangency occurs when t = to. (Write your solution using the form (*.*.*). Use t for the parameter that takes all real values. Simplify all trigonometric expressions by evaluating them. Express numbers in exact form. Use symbolic notation and fractions as needed.) 1(t) = (-5.-51.49² +14 Where will this line intersect the xy-plane? (Write your solution using the form (*.*.*). Express numbers in exact form. Use symbolic notation and fractions where needed.) point of intersection: Question

Answers

The equation of the line in the xy-plane is given by z = 0.Substituting z = 0 in the equation of the tangent line, we get:5 sin(7) - 5t = 0t = sin(7)Hence the point of intersection is given by:(5 cos(7), 0, 0)

The position vector for a particle moving on a helix is c(t)

= (5 cos(t), 5 sin(t), 12). Find the speed s(t₀) of the particle at time t₀

= 7.The position vector of the particle moving on the helix is given by:c(t)

= (5 cos(t), 5 sin(t), 12)Speed of the particle is given by:s(t)

=  |c'(t)|where c'(t) is the derivative of c(t).Differentiating the equation of the helix with respect to time we have:c'(t) = (-5 sin(t), 5 cos(t), 0)Substituting t₀

= 7:s(t₀)

= |(-5 sin(7), 5 cos(7), 0)|

= |-5 sin(7)|² + |5 cos(7)|²

= 25 Therefore the speed of the particle is 25.Find the parametrization for the tangent line at time t₀

= 7π. Use the equation of the tangent line such that the point of tangency occurs when t

= t₀. (Write your solution using the form (*.*.*). Use t for the parameter that takes all real values. Simplify all trigonometric expressions by evaluating them. Express numbers in exact form. Use symbolic notation and fractions as needed.)At t

= t₀, the point on the helix is P

= c(t₀)

= (5 cos(7), 5 sin(7), 12).Let Q be any other point on the tangent line. Let Q

= (x, y, z).Then the vector QP lies on the tangent line and is a scalar multiple of the tangent vector c'(t₀).Thus,QP

= k c'(t₀)where k is any scalar.Substituting t₀

= 7π and c'(t₀)

= (-5 sin(7π), 5 cos(7π), 0)

= (0, -5, 0)we have,QP = k(0, -5, 0)

= (0, -5k, 0)Since Q lies on the line, Q satisfies the equation of the line that is given by:(x, y, z)

= (5 cos(7), 5 sin(7), 12) + t (0, -5, 0)where t is any scalar.Substituting the value of Q, we have:(x, y, z)

= (5 cos(7), 5 sin(7) - 5kt, 12)Thus, the parametrization of the tangent line at t

= 7π is given by:(x, y, z)

= (5 cos(7), 5 sin(7) - 5t, 12)where t is any scalar.Where will this line intersect the xy-plane? (Write your solution using the form (*.*.*). Express numbers in exact form. Use symbolic notation and fractions where needed.).The equation of the line in the xy-plane is given by z

= 0.Substituting z

= 0 in the equation of the tangent line, we get:5 sin(7) - 5t

= 0t

= sin(7)Hence the point of intersection is given by:(5 cos(7), 0, 0)

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Suppose that a set of standardized test scores is normally distributed with a mean of μ=79 and standard deviation σ=8. Use the first five terms of the Maclaurin series for e−a2/2 to estimate the probability that a random test score is between 63 and 79. Round your answer to four decimal places. Provide your answer below: Probabtity

Answers

Using the first five terms of the Maclaurin series for [tex]e^(-a^2/2)[/tex], The estimated probability that a random test score is between 63 and 79 is 0.3647.

To estimate the probability, we need to calculate the standard score (z-score) for the lower and upper bounds of the range and then use the Maclaurin series approximation for the cumulative distribution function of the standard normal distribution.

The z-score for a test score of 63 is calculated as follows:

z1 = (63 - μ) / σ = (63 - 79) / 8 = -2

Similarly, the z-score for a test score of 79 is calculated as:

z2 = (79 - μ) / σ = (79 - 79) / 8 = 0

Using the Maclaurin series for [tex]e^(-a^2/2)[/tex], we can approximate the cumulative distribution function for the standard normal distribution. Taking the first five terms of the series, the approximation is:

P(63 ≤ X ≤ 79) ≈ Φ(z2) - Φ(z1)

             ≈ [tex]e^(-0^2/2)/2 - e^(-2^2/2)/2[/tex]

             ≈ 0.5000 - 0.1353

             ≈ 0.3647

Therefore, the estimated probability that a random test score is between 63 and 79 is approximately 0.3647, rounded to four decimal places.

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4. Prove that 30∈/A, where A={x∣x is integer and x=3k+5, where k is integer } 5. Prove that A⊂B. Given A={x∣x=2k+5, where k∈I+},B={x∣x=2j+1 where j∈I+} 6. From 5, prove that B⊂A. 7. Given A={x∣x=4j−5, where j∈1+and j≥2} B={x∣x=2k+1, where k∈I+and k≥0}, prove that A⊂B. 8. Given A={x∣x=2k−3, where k∈1+} B={x∣x=j+3, where j∈I+}, prove that A⊂. 9. Given A={x∈R∣x2+x−2=0} B={0,1,2,…}, prove that A=B. 10. Given A={x∈1+∣x=4j−3, where j∈I+} B={x∈1+∣x=2k−3, where k∈I+}, prove that A=B. 11. Given A={x∈1+∣x is divisible by 2} B={x∈1+∣x is divisible by 3} C={x∈1+∣x is divisible by 6}, prove the followings: 11.1 A⊂B

Answers

We can conclude that A is not a subset of B (A ⊄ B).

We have,

To prove that A ⊂ B, we need to show that every element in A is also an element of B.

In other words, if x is in A, then x must be in B.

Given:

A = {x ∈ 1+ | x is divisible by 2}

B = {x ∈ 1+ | x is divisible by 3}

Let's take an arbitrary element, say y, from A.

Since y is divisible by 2, it must be an even number.

However, not all even numbers are divisible by 3.

Therefore, y may or may not be an element of B.

This means that there exist elements in A that are not in B.

Thus,

We can conclude that A is not a subset of B (A ⊄ B).

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The complete question:

Prove that A ⊂ B, we need to show that every element in A is also an element of B. In other words, if x is in A, then x must be in B.

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Find the area of the region lying to the right of x = 2y - 6 and to the left of x = 30-2y. (Use symbolic notation and fractions where needed.) A = Steel forms will be used to cast a 12-inch-thick wall in cold weather with concrete containine 300 ular ong Type 1 cement. The wall will be wrapped with a 2-inch-thick blanket made with mineral fiber insulations. What is the minimum acceptable surrounding ambient temperature for 3 days curing without providing additional protection? Compare your response to the sample. Check all ideas that you included. the definition of commensalism the benefits to the plants from the worms the lack of observed harm to the worms the lack of observed benefit to the worms A rocket is fired at sea level and climbs at a constant angle of \( 85^{\circ} \) through a distance of 30,000 feet. Approximate its altitude to the nearest foot. \( d= \) \& \( \mathrm{ft} \) Unit operating expenses for an item costing $49 are estimated at 30% of cost, and the desired operating profit is 10% of cost. a. Determine the selling price. (Do not round intermediate calculations and round your final answer to 2 decimal places.) Selling price $ b. Determine the rate of markup on cost. (Do not round intermediate calculations and round your final answer to 1 decimal place.) Markup on cost % c. Determine the rate of markup on selling price. (Do not round intermediate calculations and round your final answer to 1 decimal place.) Markup on seling price % A giant beach ball is deflating at a rate of 3 cubic inches per minute. Find the rate at which the radius is decreasing when the ball's radius is 2 feet. FULL CREDIT WILL NOT BE GIVEN WITHOUT SUPPORTING WORK! V = r The half-life of a radioactive element is 1,200 years. How old is the object if only 12.5% of the radioactive atoms remain? Determine the total yield of ATP from the complete oxidation of palimitic acid, a 16-C saturated fatty acid. Show your work. when does valve overlap occur in the operation of an aircraft reciprocating engine? group of answer choices at the end of the exhaust stroke and the beginning of the intake stroke. at the end of the power stroke and the beginning of the exhaust stroke. at the end of the compression stroke and the beginning of the power stroke. If 8.2mol of hydrogen is produced, how many moles of hydrofluoric acid (HF) reacted? (Round to the nearest tenth)[tex]Si + 4HF -\ \textgreater \ 2H2 + SiF4[/tex] What mass of water in grams contains 1.3 g of Ca? (1.3 g of Ca is the recommended daily allowance of calcium for 19 - to 24yearolds) Express your answer in grams to two significant figures. X Incorrect; Try Again; 5 attempts remaining will the allele frequencies in the breeding pens tend to move toward the frequency in the larger field population? yes, because the breeding pen populations were established from the field population. no, because the breeding pen populations are now independent of the field population. Write a summary of the principles behind synthetic aperture radar (SAR) imaging. It will need to explain how the technique allows scatterers to be resolved along both range and cross range dimensions. Suppose that \( A \subset \mathbb{R} \) is - bounded above, - non-empty, and - \( x>0 \) for all \( x \in A \). Prove: If \( B=\left\{x^{2}: x \in A\right\} \), then \( \sup (B)=(\sup (A))^{2} \). federal agencies must specify their needs in such a manner as to: group of answer choices promote full and open competition minimize the lead-time to acquire the desired goods or services minimize cost of the procurement reduce overhead in the procurement function alpha corp. is owned by omega partners, in which abe, ben and cathy each have a one-third interest. abe's wife also owns 100% of beta corp.under the attribution rules of irc 318:how much stock in alpha corp. is beta corp. deemed to own? I need help with this Selkirk Company obtained a $36,000 note receivable from a customer on January 1, 2021. The note, along with interest at 10%, is due on July 1, 2021. On February 28, 2021, Selkirk discounted the note at Unionville Bank. The banks discount rate is 12%.Required:Prepare the journal entries required on February 28, 2021, to accrue interest and to record the discounting for Selkirk. Assume that the discounting is accounted for as a sale. Suppose f(x) is continuous on the interval [4,8], but not differentiable at x=5 and x=7. Then f(x) still achieves an absolute maximum value and an absolute minimum value along the interval [4,8]. True False a) For the parabola:f(x) = x + 4x + 4. Find the following: 1) Axis of symmetry. 2) Vertex. 3) Range. b) Differentiate the function f(x) c) Find (12x+ + 5x + 20e5x)dx = x+2 x+1