The step in showing that[tex]n^3+2n[/tex] is divisible by 3 by mathematical induction is: Prove that the statement is true when n = k. Option B
How to determine the step showing that n^3+2n is divisible by 3 is true by mathematical inductionThis step involves substituting k for n in the expression [tex]n^3+2n[/tex] and showing that the result is divisible by 3. In other words, we need to show that:
[tex]k^3 + 2k[/tex] is divisible by 3.
If we assume that the statement is true for n = k, then we can substitute k+1 for n in the expression [tex]n^3+2n[/tex] and simplify:
=[tex](k+1)^3 + 2(k+1) = k^3 + 3k^2 + 3k + 1 + 2k + 2[/tex]
= [tex]k^3 + 3k^2 + 5k + 3[/tex]
= [tex](k^3 + 2k) + (3k^2 + 3k)[/tex]
Since we assumed that the statement is true for n = k, we know that [tex]k^3 + 2k[/tex] is divisible by 3. Also, [tex]3k^2 + 3k[/tex] is divisible by 3 because it has a factor of 3. Therefore, the entire expression [tex](k+1)^3 + 2(k+1)[/tex]is also divisible by 3.
This completes the inductive step, and we can now conclude by mathematical induction that the statement [tex]n^3+2n[/tex] is divisible by 3 is true for all positive integers n.
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The price of a CD radio cassette has been reduced from $16,999 to $13,999 . Calculate the rate of discount
The rate of discount for the CD radio cassette is 17.65%.
To calculate the rate of discount, we need to find the difference between the original price and the discounted price, and then divide that difference by the original price.
Original price: $16,999
Discounted price: $13,999
So, the Discount amount
= Original price - Discounted price
= $16,999 - $13,999
= $3,000
Rate of discount = (Discount amount / Original price) * 100%
= ($3,000 / $16,999) x 100%
≈ 17.65%
Therefore, the rate of discount for the CD radio cassette is 17.65%.
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What multiplies to 14 and adds to -9
The numbers which multiplies each other to give 14 and adds to give -9 is -2 and -7
How to find factors of a number?What are the factors of 14
14 = 1, 2, 7, 14
What two factors of 14 multiplies each other to give 14 and add each other to give -9
1 × 2 = 2
1 × 7 = 7
1 × 14 = 14 ; 1 + 14 = 15
2 × 7 = 14 ; 2 + 7 = 9
So,
-2 + - 7 = -9
Therefore, negative two plus negative seven multiplies to give 14 and adds to give -9
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A giant golf ball is being constructed for the final hole of a new mini-golf course. They used 38. 7 cubic feet of material to create the figure. What is the radius of the gaint gold ball?
2. 1
38
114
9. 1
The radius of the golf ball is given as follows:
2.1 feet.
What is the volume of an sphere?The volume of an sphere of radius r is given by the multiplication of 4π by the radius cubed and divided by 3, hence the equation is presented as follows:
[tex]V = \frac{4\pi r^3}{3}[/tex]
The volume of the ball in this problem is given as follows:
V = 38.7 cubic feet.
Hence the radius of the ball is obtained as follows:
r³ = 3 x 38.7/4π
r³ = 9.2389
[tex]r = \sqrt[3]{9.2389}[/tex]
r = 2.1 feet
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a piece of rock has a mass of 86.3g , correct to the nearest 0.1g . find the greatest possible mass of the rock in 2 decimal places
The greatest possible mass of the rock to two decimal places is 86.34 g
How to find the greatest possible mass of the rock?Since the mass of the piece of rock is 86.3 g, correct to the nearest 0.1 g. Thus, the actual mass of the rock in 2 decimal places could be between 86.25 g and 86.34 g.
To find the greatest possible mass of the rock, we need to round up to the higher end of the range of the mass of the rock.
Therefore, the greatest possible mass of the rock to two decimal places is 86.34 g
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for the function y=-1+6 cos(2 pi/7(x-5)) what is the maximum value
the maximum value of the function is y = -1 + 6cos(2π/7(26.75-5)) = 5.
The function y = -1 + 6cos(2π/7(x-5)) is a periodic function with a period of 7. The maximum value of the function occurs when the cosine function reaches its maximum value of 1.
So, we need to find the value of x that makes the argument of the cosine function equal to an odd multiple of π/2, which is when the cosine function is equal to 1.
2π/7(x-5) = (2n + 1)π/2, where n is an integer
Simplifying this equation, we get:
x - 5 = (7/4)(2n + 1)
x = 5 + (7/4)(2n + 1)
Since the function has a period of 7, we can restrict our attention to the interval [5, 12].
For n = 0, we get x = 5 + 7/4 = 23/4
For n = 1, we get x = 5 + (7/4)(3) = 26.75
For n = 2, we get x = 5 + (7/4)(5) = 33.25
For n = -1, we get x = 5 + (7/4)(-1) = 1.75
For n = -2, we get x = 5 + (7/4)(-3) = -4.75
Out of these values of x, the only one that lies in the interval [5, 12] is x = 26.75.
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Light bulbs are labeled with the number of watts of electricity they use as shown by the table. What are the attributes, measurements used, and observations of the data?
The attributes, measurements used, and observations of the data are listed below:
The attributes, measurements used, and observations of the dataAttributes:
- Light bulb
- Wattage (number of watts of electricity used)
Measurements used:
- Watts (units of measurement for the amount of electricity used by a light bulb)
Observations:
- The data shows the wattage for different types of light bulbs.
- The wattage varies widely among different types of light bulbs.
- Incandescent bulbs have the highest wattage, followed by halogen bulbs, fluorescent bulbs, and LED bulbs.
- The wattage for each type of bulb can vary within a certain range, with some bulbs having higher or lower wattage than others within the same category.
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Answer:
Step-by-step explanation:
simplify the following
[tex] \frac{5 {ab } ^{2} + 10 {a}^{4} {b}^{3} - 15 {ab}^{3} }{5ab} [/tex]
a tree was 59 inches tall. two years later it was 89 inches tall. what was the percent increase, to the nearest hundredth, of the height of the tree? enter the answer in the box.
Answer:
50.85% increase
Step-by-step explanation:
The formula for % change is (new-old)/old.
So (89-59)/59 = 30/59 = 0.5084745763 = 50.85% increase
14. The diameter of a cylinder is 7 feet, and the height is 12 feet. What is the volume of the cylinder in terms of л? Recall that the volume formula for a cylinder is V = r²h.
Answer:
The formula is V = r²h for the volume of a cylinder, we need to first find the radius of the cylinder:
The diameter of the cylinder is 7 feet, so the radius is half of that cylinder, so you have to divide by 2 to get half of the diameter:
r = 7/2
= 3.5 feet.
V = π(3.5)²(12)
V = π(12.25)(12)
V = 147π
Therefore, the volume of the cylinder is 147π cubic feet.
Question 5
A player randomly selects a marble from a bag. The bag has 5 blue marbles, 6 red, 1 orange, and 1 yellow. How many outcomes are in the sample space?
The sample space has 13 outcomes.
We have to given that;
A player randomly selects a marble from a bag.
And, In the bag which has 5 blue marbles, 6 red, 1 orange, and 1 yellow.
Since, The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
Now, We can formulate;
Total number of marbles = 5 + 6 + 1 + 1
Total number of marbles = 13
Hence, there are 13 possible outcomes for the player's selection.
Therefore, We get;
The sample space has 13 outcomes.
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A 95% confidence set for two or more coefficients is a set that contains: A) the sample values of these coefficients in 95% of randomly drawn samples. B) integer values only. C) the same values as the 95% confidence intervals constructed for the coefficients. D) the population values of these coefficients in 95% of randomly drawn samples.
A 95% confidence set for two or more coefficients is a set that contains: D) the population values of these coefficients in 95% of randomly drawn samples.
A 95% confidence set for two or more coefficients is a set that contains the same values as the 95% confidence intervals constructed for the coefficients. This means that the set includes all possible values for the coefficients that are consistent with the observed data and the level of uncertainty indicated by the confidence level. It is important to note that the confidence set is not guaranteed to contain the true population values of the coefficients in 95% of randomly drawn samples, as the true values are unknown.
The confidence set provides a range of plausible values that can be used for inference and decision making. The concept of a 95% confidence set.
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William asked 20 of his friends if they own a car and wrote a report on his
findings.
The headline of his report was "65% of people do not own a car".
a) How many of the people William asked did not own a car?
b) Write a sentence to describe a problem with William's headline and a
problem with his survey.
(a) The people that did not own a car are 13
(b) The sentence to describe the problem is "13 of every 20 people do not own a car"
(a) People that did not own a carGiven that
Number of friends = 20
Percentage that do not own a car = 65%
Using the above as a guide, we have the following:
People = 65% * 20
Evaluate
People = 13
(b) The sentence to describe the problemWriting a sentence to describe the problem, we have:
The statement is given as
65% of people do not own a car".
This can be rewritten as
"13 of every 20 people do not own a car"
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Nina's car exponentially depreciates at a rate of 8% per year. If Nina bought the car when it was 7 years old for $14,000, what was the original price of the car?
Answer:
And solving for the initial amount we got:
So then the original price for the car would be approximately 18263.539$
Step-by-step explanation:
For this case we can use the exponential model given by:
Where:
represent the initial amount for the car
represent the exponential growth/decay rate
t represent the number of years. With t =0 at the begin
After 6 year we have that t =6, and we know this condition:
So then we can use the exponential model formula and the condition given and we have this:
And solving for the initial amount we got:
So then the original price for the car would be approximately 18263.539$
DUE TOMORROW!!! PLS HELP!! 1 POINT!!!
What is the measure of the unknown angle? (1 point)
image of a one hundred ninety eight degree angle and an unknown angle
a
152°
b
157°
c
162°
d
169°
Answer:
162 Degrees
Step-by-step explanation:
Think of this as a circle.
If you subtract 360 from 198, you get 162.
Which would be your missing angle.
11. A company is selling two new products. Weekly sales of product one can be
modeled by the function f(x) = 100(1.108)* after x weeks. Weekly sales of product
two can be modeled by the function g(x) = 10x + 200 after x weeks. After
approximately how many weeks will the weekly sales of the two products be the
same? (Hint: Where do they intersect)
Tide
A. 9 weeks
B. 11 weeks
C. 15 weeks
D. 19 weeks
(-2,-2) multiplied by the scale factor of 2
ANSWER:
(-4,-4)
EXPLANATION:
Just simply multiply the two points by 2. Since -2*2 is equal to -4, then the answer would be (-4, -4).
tom has 5 books with red covers and 5 books with white covers to place on a shelf. in how many ways can he arrange the books if the are to alternate red, white, red, white?
By probability , Tom can arrange his 5 red and 5 white books in an alternating pattern in 48 ways
There are 5 red books and 5 white books to arrange in an alternating pattern on the shelf.
To do this, Tom can start with either a red or a white book. If he starts with a red book, he can arrange the rest of the books in 4! (4 factorial) ways. Similarly, if he starts with a white book, he can arrange the rest of the books in 4! ways. Therefore, the total number of ways Tom can arrange the books in an alternating pattern is 2 x 4! = 48.
Tom can arrange the 10 books in an alternating pattern by starting with either a red or a white book. If he starts with a red book, he can arrange the rest of the books in 4! ways, and if he starts with a white book, he can also arrange the rest of the books in 4! ways. Therefore, the total number of ways to arrange the books is 2 x 4! = 48.
Tom can arrange his 5 red and 5 white books in an alternating pattern in 48 ways.
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The sum of the measures of the angles of a triangle is 180 m
The sum of the measures of the angle of a triangle is 180 degrees.
Sum of angles in a triangleThe given triangle is a type of triangle and the type of triangle a scalene triangle.
For a scalene triangle, the measure of the three sides are unequal and the sum of the interior angle of a triangle is 180 degrees.
Hence the measure of the angles <A, <B and <C are all less than 90 degrees since they are all acute angles.
We can therefore conclude that:
m<A + m<B + m<C = 180 degrees
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Chelsea bought a bond with a face value of $5,000. The bond has a term of 4 years. Chelsea bought the bond at a 3 percent discount from the face value. The bond pays 3 percent annual interest, and Chelsea will receive eight semiannual payments. When the bond is redeemed at maturity, the total return (profit) will be decimal place.)
Chelsea's total return (profit) when the bond is redeemed at maturity is $50.
Chelsea paid for the bond with the 3% discount:
Discount = 3% of $5,000
= $150
Price paid for the bond = $5,000 - $150
= $4,850
Next, let's calculate how much Chelsea will receive in each semiannual payment:
Semiannual interest rate = 3% / 2
= 1.5%
Semiannual payment = Face value x Semiannual interest rate = $5,000 x 1.5% = $75
Since there are eight semiannual payments, Chelsea will receive a total of:
Total semiannual payments = Semiannual payment x Number of semiannual payments = $75 x 8 = $600
Finally, let's calculate the total return (profit) when the bond is redeemed at maturity:
Total interest earned = Annual interest rate x Term x Face value = 3% x 4 x $5,000 = $600
Face value at maturity = $5,000
Total return = Price paid + Total interest earned + Total semiannual payments - Face value at maturity
Total return = $4,850 + $600 + $600 - $5,000 = $50
Therefore, Chelsea's total return (profit) when the bond is redeemed at maturity is $50.
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Solve for y.
-5(-5y+2)-8y=3(y-7)-7
Simplify your answer as much as possible.
Question 3 of 15
Answer:
y=-[tex]\frac{9}{7}[/tex]
Step-by-step explanation:
Ap calculus test. College board
Multivariable calculus in college is generally more challenging than AP Calculus BC due to the additional complexity of working with functions of several variables.
Multivariable calculus in college typically goes beyond the content covered in AP Calculus BC in high school. While both cover calculus concepts, multivariable calculus adds the complexity of working with functions of several variables, including partial derivatives, multiple integrals, and vector calculus. As a result, the difficulty level of multivariable calculus in college can be significantly higher than AP Calculus BC. However, this can vary depending on the individual student and the rigor of their high school AP course.
AP Calculus AB and BC are two levels of Advanced Placement calculus courses typically offered in high school.
AP Calculus AB covers a wide range of calculus topics including limits, derivatives, integrals, applications of derivatives and integrals, and the Fundamental Theorem of Calculus. The focus is on single-variable calculus, meaning students work with functions of one variable.
AP Calculus BC, on the other hand, covers all the topics included in AB as well as additional concepts such as parametric, polar, and vector functions, series, and sequences. The course is designed to be more challenging and covers more advanced topics in calculus, including applications of calculus in physics and engineering.
Overall, AP Calculus BC is considered to be a more rigorous course than AP Calculus AB, but both courses prepare students for college-level calculus.
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complete question:
how does multivariable calculus in college compare with ap calculus bc in high school? is there a significant gap in difficulty?
what’s the area of this figure, rounded to the nearest tenth?
Answer:
198 in²
Step-by-step explanation:
triangle (A=1/2)bh [A=1/2(22×8)] b=22 H=8
rectangle (A=bh) [A= 22×5] B=22 H=5
triangle = 88
rectangle = 110
So, 88+110 = 198
janelys has 62 m of fencing to build a four-sided fence around a rectangular plot of land. the area of the land is 220 square meters. solve for the dimensions (length and width) of the field.
Therefore, the dimensions of the rectangular plot are either 20 meters by 11 meters or 11 meters by 20 meters.
Let's assume that the length of the rectangular plot is x meters. Then the width would be (220/x) meters because we know that the area is 220 square meters.
The perimeter of the rectangular plot would be:
2x + 2(220/x) = 62
Simplifying this equation, we get:
2x^2 + 440 = 62
2x^2 - 62x + 440 = 0
x^2 - 31x + 220 = 0
Factoring this equation, we get:
(x - 20)(x - 11) = 0
Therefore, x = 20 or x = 11.
If x = 20, then the width would be 11 and if x = 11, then the width would be 20.
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Let f (x) = x^2.
A) which is greater: f(7) or f (-7)? Explain.
B) which is greater f(-10) or f(3)? Explain.
Evaluating the function we can see that:
a) f(7) is equal to f(-7)
b) f(-10) is greater than f(3).
How to see which value is greater?Here we have the quadratic function f(x) = x²
A) First we need to evaluate the function in x = 7 and x = -7 to see which one gives a larger value.
f(7) = 7² = 49
f(-7) = (-7)² = 49
Then neither of these two is greater, because are the same value.
B) now we need to evaluate in x = -10 and x = 3
f(-10) = (-10)² = 100
f(3) = 3² = 9
We can see that f(-10) is greater.
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Unit 7
1. For each equation, find the initial value and the percent increase or decrease.
a. f(x) = 37(1.04)*
i. IV:
ii. % Inc/Dec:_
b. f(x) = (-0.7) (0.6)*
i. IV:
ii. % Inc/Dec:___
Please help me
a.
i. The initial value (IV) is 37.
ii. The percent increase or decrease is 4%.
b.
i. The initial value (IV) of the function is -0.7.
ii. The percent decrease is 40%.
a.
i. The initial value (IV) of the function is 37.
ii. The percent increase or decrease is 4%, because the base of the exponential function is 1.04, which is 1 plus 4% (0.04).
b.
i. The initial value (IV) of the function is -0.7.
ii. The percent decrease is 40%, because the base of the exponential function is 0.6, which is 60% (0.6) of the initial value (-0.7).
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The complete question:
1. For each equation, find the initial value and the percent increase or decrease.
a. f(x) = 37(1.04)ˣ
i. IV:__
ii. % Inc/Dec:_
b. f(x) = (-0.7) (0.6)ˣ
i. IV:
ii. % Inc/Dec:___
For the month of January in a certain​ city, 76​% of the days are cloudy. Also in the month of January in the same​ city, 32​%
of the days are cloudy and snowy. What is the probability that a randomly selected day in January will be snowy if it is cloudy​?
The probability that a randomly selected day in January will be snowy, given that it is cloudy, is approximately 0.421 or 42.1%.
To find the probability that a randomly selected day in January will be snowy given that it is cloudy, we need to use conditional probability.
Let's denote:
C = event that a randomly selected day is cloudy
S = event that a randomly selected day is snowy
We are given:
P(C) = 76% = 0.76 (probability of a day being cloudy)
P(C and S) = 32% = 0.32 (probability of a day being both cloudy and snowy)
The conditional probability of S given C can be calculated using the formula:
P(S|C) = P(C and S) / P(C)
Plugging in the values, we have:
P(S|C) = 0.32 / 0.76 ≈ 0.421 (rounded to three decimal places)
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work out the value of x
Answer:
20
Step-by-step explanation:
4x + 5x = 180
9x = 180
x = 20
f(x)=4+3x-x^2 f(3+h)-f(3)/h
The final expression for the given problem is (f(3+h) - f(3))/h = -h - 3.
To solve this problem, we first need to determine the value of f(3+h) and f(3):
Given [tex]F(x) = 4 + 3x - x^2[/tex], we can find f(3+h) by substituting 3+h for x:
[tex]f(3+h) = 4 + 3(3+h) - (3+h)^2\\f(3+h) = 4 + 9 + 3h - (9 + 6h + h^2)\\f(3+h) = -h^2 - 3h + 4[/tex]
Next, we can find f(3) by substituting 3 for x:
[tex]f(3) = 4 + 3(3) - (3)^2\\f(3) = 4 + 9 - 9\\f(3) = 4[/tex]
Now we can substitute these values into the expression (f(3+h) - f(3))/h:
[tex](f(3+h) - f(3))/h = ((-h^2 - 3h + 4) - 4)/h\\(f(3+h) - f(3))/h = (-h^2 - 3h)/h\\(f(3+h) - f(3))/h = -h - 3[/tex]
Therefore, the final expression for the given problem is:
(f(3+h) - f(3))/h = -h - 3
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In a secondary school class, 23 students study economics 48 students study either economics or government or both (I) what is the total number of students study economics (I) how many students study government only
The answer for (1) is 31, the answer for (2) is 25, and the answer for (3) is 17.
It is given that:
In a secondary school class, 23 pupils study Economics, 6 pupils study both Government and Economics. 48 pupils study either Government or Economics or both.
Based on the information we can draw a Venn diagram:
From the Venn diagram:
A + B = 17 + 6
A + B = 23
B = 6
A + B + C = 48
A = 23 - 6 = 17
C = 48 - 17 - 6
C = 25
The total number of pupils who only study Government = 25
The total number of pupils who study Government = 25 + 6 = 31
The total number of pupils who study Economics only = 17
Thus, the answer for (1) is 31, the answer for (2) is 25, and the answer for (3) is 17.
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suppose an investor has a long box spread using the june 50 and 55 options. what is the cost of the box spread?
To determine the cost of a box spread, you need to calculate the net cost or net debit of the options involved in the spread. In a long box spread, you buy a bull call spread and a bear put spread using the same expiration date and strike prices.
In this case, the investor has a long box spread using the June 50 and 55 options. The bull call spread involves buying the June 50 call option and selling the June 55 call option, while the bear put spread involves buying the June 55 put option and selling the June 50 put option.
To calculate the cost of the box spread, you would add the cost of the bull call spread (net debit) to the cost of the bear put spread (net debit). The net debit is the difference in premiums between the options involved in each spread.
Since the specific premium values for the options are not provided in the question, it is not possible to determine the exact cost of the box spread without that information. You would need the premium values to calculate the net debit for each spread and then sum them to determine the total cost of the box spread.
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