find the area of an equilateral triangle ABC, whose sides are 7cm each and the height 4.2cm​

Answers

Answer 1

Step-by-step explanation:

Area of equalaterial triangle is 1/2bh

Here the base is 7.

The height is 4.2

[tex] \frac{1}{2} (7)(4.2) = 14.7[/tex]


Related Questions

Which inequality has infinitely many negative integer solutions?

x>5
0 < x
x ≤ 1
x ≥8

Answers

The inequality "x ≤ 1" has infinitely many negative integer solutions as any negative integer value for x satisfies this condition.

The inequality that has infinitely many negative integer solutions is "x ≤ 1".

To understand why this is the case, let's break down the inequality and consider its implications. The inequality x ≤ 1 means that x is less than or equal to 1. In other words, any value of x that is smaller than or equal to 1 satisfies this inequality.

Now, when it comes to negative integers, they are numbers less than zero. Since negative integers are smaller than 1, any negative integer value for x will fulfill the condition x ≤ 1. For example, if we take x = -1, -2, -3, and so on, they are all less than or equal to 1, thereby satisfying the inequality.

Moreover, there are infinitely many negative integers (-1, -2, -3, and so on), meaning that there is no limit to the number of negative integer solutions that satisfy x ≤ 1. Hence, the inequality has infinitely many negative integer solutions.

On the other hand, the inequalities "x > 5" and "0 < x" do not have infinitely many negative integer solutions. "x > 5" implies that x must be greater than 5, excluding negative integers. "0 < x" states that x must be greater than zero, also excluding negative integers. Finally, "x ≥ 8" requires x to be greater than or equal to 8, which again does not include negative integers.

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standard form of x^2 - 3 = 4x^2 + 6

Answers

Answer: x^4 - 2x^2 - 24

Question 9
The volume of a cube can be found using the equation V = s³, where V is the volume and s is the measure of one side of the cube.
Match the equation for how to solve for the side length of a cube to its description.
Drag the equation into the box to match the description.

Answers

The side length of the cube is [tex]s = \sqrt[3]{V}[/tex]

How to solve for the side length of the cube

From the question, we have the following parameters that can be used in our computation:

V = s³

Where

V is the volumes is the measure of one side of the cube.

Using the above as a guide, we have the following:

V = s³

Take the cube root of both sides

[tex]s = \sqrt[3]{V}[/tex]

Hence, the side length of the cube is [tex]s = \sqrt[3]{V}[/tex]

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A timer is started and a few moments later a swimmer dives into the water and then comes back up. The swimmer's depth (in feet) as a function of time (in seconds after the timer was started) is given by the equation h(t)=t^2−16t+55


Rewrite the formula in factored form and select each true statement below.



The swimmer reaches a maximum depth of 16 feet.

The swimmer reaches a maximum depth of 16 feet.

The swimmer dives into the water 5 seconds after the timer is started.

The swimmer dives into the water 5 seconds after the timer is started.

The swimmer comes back up 16 seconds after the timer is started.

The swimmer comes back up 16 seconds after the timer is started.

The swimmer comes back up 11 seconds after the timer is started.

The swimmer comes back up 11 seconds after the timer is started.

This function shows that the swimmer dives into the water two separate times.

(sorry the answers duplicated themselves)

Answers

Answer: The formula in factored form is h(t) = (t - 5)(t - 11). The true statements are B and D.

Step-by-step explanation: To rewrite the formula in factored form, we need to find two numbers that multiply to 55 and add to -16. These numbers are -5 and -11. So, we can write:

h(t) = t^2 - 16t + 55 h(t) = (t - 5)(t - 11)

This means that the swimmer’s depth is zero when t = 5 or t = 11. These are the times when the swimmer dives into the water and comes back up, respectively. So statement B and D are true.

To find the maximum depth of the swimmer, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by:

x = -b/2a x = -(-16)/2(1) x = 8

The y-coordinate of the vertex is given by:

y = h(8) y = (8 - 5)(8 - 11) y = (-3)(-3) y = 9

So the maximum depth of the swimmer is 9 feet, not 16 feet. Therefore, statement A is false.

Statement C is also false because the swimmer comes back up at t = 11, not t = 16.

Statement E is also false because the function shows that the swimmer dives into the water only once, not twice. The function has only two zeros, which correspond to the times when the swimmer enters and exits the water.

Hope this helps, and have a great day! =)

 Miguel deposited a certain amount of money in the bank. The bank paid him interest after one year at which point he had $757.12. After the next year he had $787.40. How much money did Miguel originally put into the bank? (Answer to the nearest dollar.)

Answers

Miguel originally put $728 into his bank account at a rate of 4% annually

What is an equation?

An equation is an expression that shows how numbers and variables are related to each other.

Let r represent the interest rate. Between successive years, Miguel money increased from $757.12 to $787.40. Hence:

Interest rate = $787.40 / $757.12 = 1.04

The interest rate = 4% annually.

Let x represent the money Miguel had initially, hence:

x * 1.04 = $757.12

x = $728

Miguel originally put $728 into his bank account

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The scatter plot below shows the number of pizzas sold during weeks when different numbers of coupons were issued. The equation represents the linear model for this data. y = 3.4x + 43 According to the model, what is the average number of pizzas sold in one night if no coupons are issued? Responses 0 pizzas 0 pizzas 21 pizzas 21 pizzas 43 pizzas 43 pizzas 60 pizzas 60 pizzas 70 pizzas

Answers

Answer: 43

Step-by-step explanation:

Look at the graph when 0 coupons are sold - the line is at about 45. The closest answer to that is 43.

Right triangle trig ratios, please answer fast, 20 points!

Answers

Step-by-step explanation:

SIn      S-O-H

sin is   opposite leg / hypotenuse

sin Φ = 20/29 =.69

Conduct a survey based on one of the topics below and write a research report. You are required to collect, represent, analyse, interpret and report the data. The topics are as follows: • The number of coins that teachers carry with them • The distance between a teacher's place of residence and her/his place of employment (school) • The school shoe sizes of grade 4 learners Follow these steps: Step 1: Collecting and organising data 1. Select one of the topics above and formulate it into a researchable title. (2) 2. Write down the main research question that will guide your investigation. (2) 3. What will be the source(s) of data for the topic you have selected? (2) 4. What data collection instrument will you use for your investigation? (2) 5. State whether you will obtain categorical data or numerical data. If the data will be numeric, state whether it will be discrete or continuous. (3) 6. Write down the question(s) that you would include in the instrument of your choice to help you get the information (data) necessary to answer the main research question. (2) 7. Gather data from a sample of 20–25 respondents using the instrument you designed. Attach one of the completed instruments to your solution. (10) 8. Use tally marks and a table to record the data in question 7. (7) [30] Step 2: Representing data 9. Which graph best represents the data that you sorted or organised in question 8? Explain your answer. (6) 10. Draw the graph that best represents the data you sorted in question 8. (5) 11. Write down two questions that will assist you in interpreting the graph that you drew in question 9.

Answers

Title: The Number of Coins That Teachers Carry with Them

Research Report

1. Research Question: What is the average number of coins carried by teachers?

2. Data Source: The data will be collected directly from teachers in a sample group.

3. Data Collection Instrument: A survey questionnaire will be used to collect the necessary data. The questionnaire will contain questions about the number of coins carried by teachers on a regular basis.

4. Data Type: Numerical data will be obtained, specifically discrete data.

5. Instrument Questions:

a) How many coins do you carry with you on a daily basis?

b) What is the highest number of coins you have carried with you?

6. Completed Instrument: Please find the attached completed survey questionnaire.

7. Data Collection: Data will be gathered from a sample of 20-25 respondents using the survey questionnaire.

8. Tally Marks and Data Table:

Number of Coins Tally

1

2

3

4

5

6

7

8

9

10

Step 2: Representing Data

9. The graph that best represents the data collected would be a bar graph. A bar graph is suitable for displaying discrete numerical data, such as the number of coins carried by teachers. It allows for easy visualization and comparison of the frequencies of different coin quantities.

Graph:

 |          

 |  |        

 |  ||    |  

 |  |||   |  

 |  ||||  |  

 |  ||||  |  

 |  ||||  |  

 |  ||||  |  

 |  ||||  |  

-----+--------+-----

1 5 10 15 20

10. Questions for Interpreting the Graph:

a) What is the most common number of coins carried by teachers?

b) How many teachers carry more than five coins with them?

11. Interpretation:

Based on the bar graph, it can be observed that the most common number of coins carried by teachers is around 4. Additionally, approximately 10 teachers carry more than five coins with them. This information provides insights into the typical coin-carrying habits of teachers within the sample group.

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Help please I need help finding the slope that is perpendicular to the given line

Answers

So from the graph we can see the slope to be -1/6. Perpendicular is the opposite slope. So this means we will flip it causing it be 6.

A coffee shop serves an average of 75 customers per hour during the morning rush. Find the probability that 80 customers arrive in an hour during tomorrow's morning rush. Round answer to 4 decimal places.

Answers

The probability that 80 customers will arrive during tomorrow's morning rush is approximately 0.0629, rounded to 4 decimal places.

To find the probability that 80 customers will arrive during tomorrow's morning rush, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space.

In this case, the average number of customers per hour is given as 75. The Poisson distribution is defined by a single parameter, λ (lambda), which represents the average rate of the event occurring in the given interval.

To calculate the probability, we can use the formula:

P(x; λ) = (e^(-λ) * λ^x) / x!

where P(x; λ) represents the probability of x events occurring given the average rate λ.

In this case, x = 80 and λ = 75. Plugging these values into the formula, we get:

P(80; 75) = (e^(-75) * 75^80) / 80!

Calculating this expression using a calculator or software, we find that P(80; 75) ≈ 0.0629.

The probability that 80 customers will arrive during tomorrow's morning rush is approximately 0.0629, rounded to 4 decimal places.

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If y=3x−2 and x=6 , find the value of y .

Answers

Answer:

y = 16

Step-by-step explanation:

y = 3x - 2 ← substitute x = 6

y = 3(6) - 2 = 18 - 2 = 16

A recent survey showed that exactly 38%
of people in a town buy the local
newspaper. There are 2450 people in the
town.
a) How many people in the town buy the
local newspaper?
b) How many people in the town do not
buy the local newspaper?

Answers

Answer:

a) .38 × 2,450 = 931 people buy the local newspaper.

b) 2,450 - 931 = 1,519 people do not buy the local newspaper.

b) In a certain weight lifting machine, a weight of 1 kN is lifted by an effort of 25 N. While the weight moves up by 100 mm, the point of application of effort moves by 8 m. Find mechanical advantage, velocity ratio and efficiency of the machine​

Answers

The mechanical advantage of the given machine is 40, Velocity ratio is 80 and efficiency is 0.5.

The given question is concerned with finding the mechanical advantage, velocity ratio and efficiency of a weight lifting machine.

The problem has provided the following information:

Weight of the object, W = 1 kN = 1000 NEffort applied, E = 25 NHeight through which the object is lifted, h = 100 mm = 0.1 m Distance through which the effort is applied, d = 8 m

We know that, mechanical advantage = load/effort = W/E and velocity ratio = distance moved by effort/distance moved by the load.Mechanical advantage

The mechanical advantage of the given machine is given by; Mechanical advantage = load/effort = W/E= 1000/25= 40Velocity ratioThe velocity ratio of the given machine is given by;

Velocity ratio = distance moved by effort/distance moved by the load.= d/h = 8/0.1= 80EfficiencyThe efficiency of the given machine is given by;

Efficiency = (load × distance moved by load) / (effort × distance moved by effort)Efficiency = (W × h) / (E × d)= (1000 × 0.1) / (25 × 8)= 0.5

Therefore, the mechanical advantage of the given machine is 40, velocity ratio is 80 and efficiency is 0.5.

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8. Paul draws a segment with the following points: (3,6) and (-4,8). He
performs a transformation and the image has points (9,18) and
(-12, 24) respectively. What type of transformation did Paul perform?

Answers

Paul performed an enlargement or dilation Transformation on the original segment to obtain the image segment.

To determine the type of transformation Paul performed, we can analyze how the coordinates of the points changed from the original segment to the image segment.

The original segment has two points: (3, 6) and (-4, 8). The image segment as two points: (9, 18) and (-12, 24).

Let's calculate the changes in the x-coordinates and y-coordinates separately:

For the x-coordinates:

- The x-coordinate of the first point changed from 3 to 9. This is an increase of 6.

- The x-coordinate of the second point changed from -4 to -12. This is also an increase of 8.

For the y-coordinates:

- The y-coordinate of the first point changed from 6 to 18. This is an increase of 12.

- The y-coordinate of the second point changed from 8 to 24. This is also an increase of 16.

From the changes observed, we can conclude that both the x-coordinates and y-coordinates increased by the same scale factor. This indicates a dilation or enlargement transformation.

Additionally, since the scale factor is greater than 1 (the coordinates increased), it suggests that the dilation was an enlargement.

Therefore, Paul performed an enlargement or dilation transformation on the original segment to obtain the image segment.

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find f(g(x)) Find g(f(x))

Answers

Answer:

f(g(x)) = x + 11

g(f(x)) = √(x² + 11)

Step-by-step explanation:

f(x) = x² + 5

g(x) = √(x + 6)

f(g(x)) = f(√(x + 6))

= (√(x + 6))² + 5

= x + 6 + 5

f(g(x)) = x + 11

g(f(x)) = g(x² + 5)

= √(x² + 5 + 6)

g(f(x)) = √(x² + 11)

What is the equation of the straight line shown below? Give your answer in the form y = mx + c, where m and c are integers or fractions in their simplest forms.
PLEASE HELP!ASAP!


(Photo given below.)​

Answers

Answer:

y = 2x + 8

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 4, 0) and (x₂, y₂ ) = (0, 8) ← 2 points on the line

m = [tex]\frac{8-0}{0-(-4)}[/tex] = [tex]\frac{8}{0+4}[/tex] = [tex]\frac{8}{4}[/tex] = 2

the line crosses the y- axis at (0, 8 ) ⇒ c = 8

y = 2x + 8 ← equation of line

An equation is shown below: 5x + 3(x- 5) =6x + 2x- 15 What is the solution to this equation?​

Answers

Answer:

All Real Number

Step-by-step explanation:

5x + 3(x - 5) = 6x + 2x - 15

5x + 3x - 15 = 8x - 15

8x - 15 = 8x - 15

-15 = -15

Since the -15 = -15, the equation will always be true for any value of x.

So, the answer is All Real Number.

Please help I need help urgently

Answers

9. How does g(x) differ from the graph of f(x): A. the graph translates to the right 1 unit.

10. How does g(x) differ from the graph of f(x): A. the graph translates up 1 unit.

What is a translation?

In Mathematics and Geometry, the translation of a graph to the right adds a number to the numerical value on the x-coordinate of the pre-image:

g(x) = f(x - N)

Conversely, the translation of a graph up adds a number to the numerical value on the y-coordinate of the pre-image:

g(x) = f(x) + N

Question 9.

The graph of the parent exponential function [tex]f(x) = 6^x[/tex] is shifted to the right by 1 unit, in order to produce the graph of this transformed exponential function [tex]g(x) = 6^{x-1}[/tex].

Question 9.

The graph of the parent exponential function [tex]f(x) = 2^x[/tex] is shifted to the up by 1 unit, in order to produce the graph of this transformed exponential function [tex]g(x) = 2^{x}+1[/tex].

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nine balls marked1 to 9 are placed in a box. if you pick one ball at random. what is the probability that an 8 is taken out

Answers

Answer:

P(8) = 1/9

Step-by-step explanation:

There are 9 balls, 1,2,3,4,5,6,7,8,9

We want to get ball 8

P(8) = number of balls marked 8/ total number of balls

       = 1/9

Answer:

P = 1/9

Step-by-step explanation:

In this scenario, there are a total of nine balls in the box, numbered from 1 to 9 (1,2,3,4,5,6,7,8,9). We are interested in finding the probability of selecting the ball marked with the number 8.The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.In this case, the number of favorable outcomes is 1 because there is only one ball marked with the number 8.The total number of possible outcomes is 9 since there are nine balls in total.

Probability of selecting the ball marked with 8

[tex]\sf Probability = \dfrac{Number\: of \:favorable\: outcomes }{ Total \:number\: of\: possible\: outcomes}\\\\\sf Probability = \dfrac{1 }{9}[/tex]

PLEASE HELP

1,0008 = 10w
W =

Answers

The simplified equation 1,0008 = 10w can be represented as 1024 = 10w, and we can determine that w is equivalent to 24 by comparing the exponents. This implies that the value of w required to make the equation true is 24.

To simplify the equation[tex]1,000^{8}[/tex]= [tex]10^{w}[/tex], we need to recognize that 1,000 is equal to [tex]10^{3}[/tex]. Therefore, we can rewrite the equation as [tex](10^{3})^{8}[/tex] = [tex]10^{w}[/tex]

Using the property of exponents, when raising a power to another power, we multiply the exponents. Applying this property, we get [tex]10^{3*8}[/tex] = [tex]10^{w}[/tex].

Simplifying further, we have [tex]10^{24}[/tex] = [tex]10^{w}[/tex].

In order for two exponential expressions to be equal, their bases must be the same. In this case, both bases are 10. Therefore, we can conclude that the exponents must also be equal.

Hence, w = 24.

The simplified equation 1,000^8 =[tex]10^{w}[/tex]can be expressed as [tex]10^{24}[/tex] = [tex]10^{w}[/tex], and by comparing the exponents, we find that w is equal to 24. This means that the value of w that makes the equation true is 24.

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When Boris had 3 years left in college, he took out a student loan for $15,485. The loan has an annual interest rate of 6.6%. Boris graduated 3 years after acquiring the loan and began repaying the loan immediately upon graduation.
According to the terms of the loan, Boris will make monthly payments for 10 years after graduation. During the 3 years he was in school and not making payments, the loan accrued simple interest.

Answers

Let's break down the calculation into several steps.

1. First, we will calculate the amount of interest that accrued over the 3 years that Boris was in school. This will be calculated using the formula for simple interest, which is:

   I = PRT,

   where:

   I is the interest,

   P is the principal amount (the initial amount of money),

   R is the rate of interest per period, and

   T is the time the money is invested for.

2. Next, we will add this accrued interest to the initial loan amount to find out how much Boris owes when he starts repaying the loan.

3. Finally, we will calculate the monthly payments Boris has to make. This will be calculated using the formula for an amortizing loan, which takes into account both the principal and the interest on the loan. The formula for the monthly payment of an amortizing loan is:

   M = P[r(1 + r)^n] / [(1 + r)^n – 1],

   where:

   M is your monthly payment,

   P is the principal loan amount,

   r is your monthly interest rate (annual interest rate divided by 12),

   n is the number of payments (the number of months you will be paying the loan).

Let's do these calculations.

1. Calculate the accrued interest over the 3 years in school:

   I = PRT

   I = $15,485 * 6.6/100 * 3

   I = $3060.57

2. Calculate the total amount owed at the start of repayment:

   Total = Principal + Interest

   Total = $15,485 + $3060.57

   Total = $18,545.57

3. Calculate the monthly payments:

   First, convert the annual interest rate to a monthly rate:

   r = 6.6% / 12 / 100 = 0.0055 per month

   Then, use the formula for the monthly payment of an amortizing loan:

   M = $18,545.57 * [0.0055(1 + 0.0055)^120] / [(1 + 0.0055)^120 – 1]

   M = $210.83

So, Boris will have to make monthly payments of approximately $210.83 for 10 years after he graduates.

Sarah is looking to take out a mortgage for $300, 000 from a bank offering a monthly
interest rate of 0.525 %. If Sarah makes monthly payments of $1925, determine how
long it will take her to pay off the loan, to the nearest tenth of a year, using the
formula below.

Answers

It will take Sarah approximately 15.9 years to pay off the loan.


To determine how long it will take Sarah to pay off the loan, we can use the formula for the number of periods required to pay off a loan:

[tex]\(n = \frac{{\log\left(\frac{{P \cdot r}}{{P - M \cdot r}}\right)}}{{\log(1 + r)}}\)[/tex]

Where:

n is the number of periods (in this case, the number of months)

P is the principal amount (loan amount), which is $300,000

r is the monthly interest rate, which is 0.525% or 0.00525 (expressed as a decimal)

M is the monthly payment, which is $1,925

Substituting the values into the formula, we have:

[tex]\(n = \frac{{\log\left(\frac{{300,000 \cdot 0.00525}}{{300,000 - 1,925 \cdot 0.00525}}\right)}}{{\log(1 + 0.00525)}}\)[/tex]

Using a calculator to evaluate the logarithms and perform the division, we find that [tex]\(n \approx 190.6\)[/tex] months.

Since we are asked to determine the time to the nearest tenth of a year, we divide 190.6 by 12 to convert it into years:

[tex]\(\frac{{190.6}}{{12}} \approx 15.9\) years[/tex]

Therefore, it will take Sarah approximately 15.9 years to pay off the loan.


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Given the diagram below, what is tan (60°)?
5
60°
30°
Triangle not drawn to scale
OA. √5
OB. 5√2
O
C. √3
OD.
.

Answers

The value of tan(60°) is √3, and option C is correct.

In order to find the value of tan(60°) using the given diagram, we need to use the tangent formula which is:tan(x) = opposite / adjacentIn the case of the given diagram, we can see that the angle we're interested in is 60°, and the side opposite the angle is the length of the line marked "√3" and the side adjacent to the angle is the length of the line marked "1".Thus, we can plug in these values into the formula:tan(60°) = opposite / adjacenttan(60°) = √3 / 1tan(60°) = √3.

The value of tan(60°) is √3, and option C is correct.

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Alexis bought x boxes of cereal for 3$each , including tax and paid with a 20$ bill which expression represents the total amount of change she should receive?
20 - 3x
(3 - 20)x
3x -20
(20 - 3)x

Answers

The expression that represents the total amount of change Alexis should receive is:

20 - 3x

Cereals are the grains belonging to the grain family. Wheat, rice, corn, millet, rye, and many others. These are rich in carbohydrates.

The cereals most commonly cultivated are wheat, rice, rye, oats, barley, corn (maize), and sorghum.

The expression that represents the total amount of change Alexis should receive is:

20 - 3x

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Answer:

A. 20 – 3x

Step-by-step explanation:

How to write in sloe intercept form

Answers

Step-by-step explanation:

Slope intercept form is

[tex]y = mx + b[/tex]

So you need to isolate the y variable

For the first equation

[tex]2x + y = - 3[/tex]

[tex]y = - 2x - 3[/tex]

For the second equation,

[tex]y = - 3 - 2x[/tex]

Notice that this system will have infinite solutions

how to solve for s in P=b+2s

Answers

To solve for s in P = b + 2s, we need to isolate s on one side of the equation.

We can do this by subtracting b from both sides of the equation:

P - b = 2s

Then, we can divide both sides by 2:

(P - b) / 2 = s

Therefore, s = (P - b) / 2.

17. Diane is making a quilt. She needs three
pieces with a yellow undertone, two pieces
with a blue undertone, and four pieces with
a white undertone. If she has six squares
with a yellow undertone, five with a blue
undertone, and eight with a white undertone
to choose from, in how many ways can she
choose the squares for the quilt?

Answers

There are 14,000 different ways for Diane to choose the squares for her quilt.

In how many ways can she choose the squares for the quilt?

we want to find the number of ways Diane can choose the squares for her quilt, we can use the concept of combinations.

Diane needs to choose three squares with a yellow undertone, two squares with a blue undertone, and four squares with a white undertone. She has six squares with a yellow undertone, five squares with a blue undertone, and eight squares with a white undertone to choose from.

The number of ways to choose the squares for each color can be calculated using combinations:

Number of ways to choose yellow squares = C(6, 3) = 6! / (3! * (6 - 3)!) = 20Number of ways to choose blue squares = C(5, 2) = 5! / (2! * (5 - 2)!) = 10Number of ways to choose white squares = C(8, 4) = 8! / (4! * (8 - 4)!) = 70

To calculate the total number of ways to choose the squares for the quilt, we multiply the number of ways for each color together:

Total number of ways = 20 * 10 * 70 = 14,000

there are 14,000 ways for Diane to choose the squares for her quilt.

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If 20% of a number is 72 and 50% of the same number is 180, find 30% of that number.

Answers

Answer:

.20x = 72, so x = 360

.30(360) = 108

You may need to use the appropriate appendix table or technology to answer this question.

Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 10 ounces.

(a) The process standard deviation is 0.16, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.84 or greater than 10.16 ounces will be classified as defects. (Round your answer to the nearest integer.)

Calculate the probability of a defect. (Round your answer to four decimal places.)

Calculate the expected number of defects for a 1,000-unit production run. (Round your answer to the nearest integer.)
defects

(b) Through process design improvements, the process standard deviation can be reduced to 0.08. Assume the process control remains the same, with weights less than 9.84 or greater than 10.16 ounces being classified as defects.

Calculate the probability of a defect. (Round your answer to four decimal places.)

Calculate the expected number of defects for a 1,000-unit production run. (Round your answer to the nearest integer.)
defects

(c) What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?

a. Reducing the process standard deviation causes a substantial reduction in the number of defects.

b. Reducing the process standard deviation causes no change in the number of defects.

c. Reducing the process standard deviation causes a substantial increase in the number of defects.

Answers

The answer to the following questions is: (a) So, the expected number of defects = 1000 × 0.3174 ≈ 317, rounded to the nearest integer. (b) expected number of defects = 1000 × 0.0456 ≈ 46, rounded to the nearest integer. (c) So, the option (a) is correct.

(a) The probability of a defect is 0.1587.  The z-values for 9.84 and 10.16 are -1.00 and 1.00 respectively. Therefore, P( -1.00 < z < 1.00) = P(z < 1.00) - P(z < -1.00) = 0.8413 - 0.1587 = 0.6826. Hence, the probability of a defect is 1- 0.6826 = 0.3174, rounded to four decimal places.
The expected number of defects for a 1,000-unit production run can be calculated by multiplying the number of units produced by the probability of defects. So, the expected number of defects = 1000 × 0.3174 ≈ 317, rounded to the nearest integer.

(b) The probability of a defect is 0.0228.  The z-values for 9.84 and 10.16 are -1.00 and 1.00 respectively. Therefore, P(-1.00 < z < 1.00) = P(z < 1.00) - P(z < -1.00) = 0.9772 - 0.0228 = 0.9544. Hence, the probability of a defect is 1 - 0.9544 = 0.0456, rounded to four decimal places.
The expected number of defects for a 1,000-unit production run can be calculated by multiplying the number of units produced by the probability of defects. So, the expected number of defects = 1000 × 0.0456 ≈ 46, rounded to the nearest integer.

(c) Reducing the process variation has an advantage in terms of quality control. If the process variation is decreased, the control limits are placed at a greater number of standard deviations from the mean. In order to detect changes in the process more quickly, the control limits must be placed at a greater distance from the mean.

By reducing process variation, the number of defective items produced can be minimized. Hence, the advantage of reducing process variation is reducing the process standard deviation causes a substantial reduction in the number of defects. Therefore, the option (a) is correct.

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Simplify i10
Enter the power that is a multiple of 4 first

Answers

i^10 = (i^2)^5

= (-1)^5

= -1
Final answer:

The mathematical term 'i' is the imaginary unit which is defined by its property i²=-1. The powers of 'i' follow a pattern that repeats every 4 numbers, so i¹⁰ is the same as i², which is -1.

Explanation:

In mathematics, i is the imaginary unit, which is defined by the property that i2 = -1. When raising i to a power, there's a pattern that repeats every 4 numbers. Specifically, i1 = i, i2 = -1, i3 = -i, and i4 = 1. This pattern starts over with i5 = i, i6 = -1, etc.

Given this, to find i10, you can realize that 10 is a multiple of 4 (2*4) plus 2. So i10 is same as i2, which is -1.

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