The function v(t) 1350(1.017)t represents the value v(t), in dollars, of a comic book t years after its purchase. the yearly rate of appreciation of the comic book is

Rewriting [tex]z[/tex] in polar form makes this trivial.

[tex]z=|z|e^{i\mathrm{arg}(z)}[/tex]

We have

[tex]|z|=\sqrt{(-1)^2+(-\sqrt3)^2}=2[/tex]

[tex]\mathrm{arg}(z)=\tan^{-1}(-1,-\sqrt3)=-\dfrac{2\pi}3[/tex]

(not to be confused with the standard inverse tangent function [tex]\tan^{-1}x[/tex]. Here [tex]\tan^{-1}(x,y)[/tex] is the inverse tangent function that takes into account position in the coordinate plane; look up "atan2" for more information)

So we have

[tex]z=-1-\sqrt3\,i=2e^{-2\pi/3\,i}[/tex]

Then

[tex]z^6=2^6\left(e^{-2\pi/3\,i}\right)^6=64e^{-4\pi\,i}=64[/tex]

so that [tex]a=64[/tex] and [tex]b=0[/tex].

Answer:

they both have the equilateral lines on each side

Step-by-step explanation:

Answer:

B. Multiply 5 times 3. Multiply 5 times 4. Add the two products

Step-by-step explanation:

Right from the start the question is only asking for squash plants so the other two rows don't need to be counted.

7-2 = 5

In each row there's 3 yellow squash = 5*3 = 15

And 4 green squash = 5*4 = 20

which means there's 35 squash total in the garden.

Answer:

$48

Step-by-step explanation:

we know that

$80 represent 100%

so

using proportion

Find how much is 60%

80/100=x/60

x=80*60/100

x=$48

Answer:

Add the five terms.

Use the formula for the fifth partial sum.

The fifth partial sum represents the total amount of money the athlete earns over the first five years.

Step-by-step explanation:

Add the five terms.

Use the formula for the fifth partial sum.

The fifth partial sum represents the total amount of money the athlete earns over the first five years.

Answer:  The fifth partial sum = 2210

The fifth partial sum represent the sum of the first 5 terms in the sequence.

Step-by-step explanation:

We know that he partial sum of a sequence gives the sum of the first n terms in the sequence.

Thus, the fifth partial sum represent the sum of the first 5 terms in the sequence.

The given geometric sequence : $400, $420, $441, $463, and $486.

Now, the fifth partial sum of the above sequence will be :

[tex]S_5=\$400+ \$420+ \$441+ \$463+\$486=\$2210[/tex]

You are given h(t) = 64–16t^2.

You need to find h(1.25).

Set up:

h(1.25) = 64 - 16(1.25)^2

Take it from here.

Answer:

Step-by-step explanation:

Given the height of a beach ball rolling off a cliff as a function of time

h = 64 — 16t²

We want to find the average change in height when t = 1.25s

The rate of change of height is the differentiation of the height I.e

∆h/∆t → dh/dt

Therefore

h = 64 — 16t².

Therefore

Differential of a constant is zero

dh/dt = —32t

At t = 1.25s

dh/dt = —32(1.25)

dh/dt = —40 ft/s

So, the height is reducing at a rate of 40ft in every seconds.

1.63

4.89 divided by 3 = 1.63

1.63 just divided the total by 3

Answer:

jump discontinuity at x = 0; point discontinuities at x = –2 and x = 8

Step-by-step explanation:

From the graph we can see that there is a whole in the graph at x=-2.

This is referred to as a point discontinuity.

Similarly, there is point discontinuity at x=8.

We can see that both one sided limits at these points are equal but the function is not defined at these points.

At x=0, there is a jump discontinuity. Both one-sided limits exist but are not equal.

The accurate discontinuities for the graph that we have here is jump discontinuity at x = 0; point discontinuities at x = –2 and x = 8.

This is the point in the graph where the values that are supposed to be in the equation would have to jump instead of being continuous. This is the point where there are broken lines.

The points in the graph that we have here can be gotten in the second option of the question.

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Answer:

10976π/3 cm³

Step-by-step explanation:

Volume of a sphere is:

V = 4/3 π r³

where r is the radius (half the diameter).

The diameter is 28 cm, so the radius is 14 cm.

V = 4/3 π (14 cm)³

V = 10976π/3 cm³

Final answer:

The volume of a basketball with a diameter of 28 cm is 4,304π cm³ when expressed in terms of π.

Explanation:

To find the volume of a basketball in terms of π (pi), we start by noting that the formula for the volume of a sphere is V = ⅔πr³, where r is the radius of the sphere. Given that the diameter of the basketball is 28 cm, we will first find the radius by dividing the diameter by 2, which gives us a radius of 14 cm. We then plug this into the formula:

V = ⅔π(14 cm)³

After simplifying, we get:

V = ⅔π(2744 cm³)

And finally, the volume V in terms of π is:

V = 4,304π cm³

Answer:

$219.6

Step-by-step explanation:

we know that

The sell price is equal to the cost price plus the profit

Let

C ----> the cost price

S----> the sell price

P ----> the profit

so

S=C+P

we have

C=$180

Find the profit

22%=22/100=0.22

P=0.22*(180)=$39.6

Find the sell price

S=$180+$39.6=$219.6

Answer:

18 Quarts

Step-by-step explanation:

there are 4 quarts per gallon

I multiplied 4 x 5 = 20 then subtracted the 2 quarts left which comes out to 18 quarts used.

Answer:

  6

Step-by-step explanation:

In order for 132* to be a multiple of 2, the digit * must be even.

In order for 132* to be a multiple of 3, the sum of all digits must be a multiple of 3. That is (1 +3 +2 + *) = 6+* is a multiple of 3. Since 6 is already a multiple of 3, then * must be a multiple of 3.

Single-digit values of * that are both even and a multiple of 3 are 0 and 6. We know that any number ending in 0 will be a multiple of 5, so we cannot use that digit.

The only possible digit that makes 132* a multiple of 2 and 3, but not 5, is 6.

Answer:

112560

Step-by-step explanation:

The total profit is the sale price of the helmet is $ 60 is:

                        $ 27,600

p denote the total profit and s denote the sale price.

The total profit in terms of the sale price is given by the function :

        [tex]p(s)=-24s^2+2200s-18000[/tex]

Now we are asked to find the value of p when the s=60

when s=60 we have:

[tex]p(60)=-24\times (60)^2+2200\times 60-18000\\\\\\p(60)=-24\times 3600+132000-18000\\\\\\p(60)=-86400+132000-18000\\\\\\p(60)=27600[/tex]

          Hence, the total profit when sale price is $ 60 is:

                              $ 27,600

Answer:

The product is equal to [tex]3\sqrt{15}[/tex]

Step-by-step explanation:

Ok,

First, you have

[tex](3\sqrt{-3})(\sqrt{-5})[/tex]

We must follow the rule for multiplying radicals, in this case:

[tex](3\sqrt[2]{-3})(\sqrt[2]{-5})=(3\sqrt[2]{(-3)(-5)})[/tex]

Note that the types of root, n, have to match, in this case is 2 for each root

Then, we know that two negative numbers multiplied give a positive number, in this case

[tex](3\sqrt[2]{-3} )(\sqrt[2]{-5})=(3\sqrt[2]{(-3)(-5)})=(3\sqrt[2]{(15)})[/tex]

So, the correct answer is [tex](3\sqrt{-3})(\sqrt{-5} )=3\sqrt{15}[/tex]

Answer:

The measure of angle 6 is m∠6=50°

Step-by-step explanation:

we know that

m∠3+m∠6=180° ----> supplementary angles (interior consecutive angles)

so

we have

m∠3=130°

substitute

130°+m∠6=180°

m∠6=180°-130°=50°

Answer:

The weight of the enlarged paperweight is [tex]1.2\ lb[/tex]

Step-by-step explanation:

we know that

If two figures are similar, then the ratio of its volumes (or its weights) is equal to the scale factor elevated to the cube

Let

z-----> the scale factor

x ----> the enlarged paperweight weight

y ----> the original paperweight weight

[tex]z^{3}=\frac{x}{y}[/tex]

we have

[tex]z=2[/tex]

[tex]y=0.15\ lb[/tex]

substitute and solve for x

[tex]2^{3}=\frac{x}{0.15}[/tex]

[tex]x=2^{3}(0.15)=1.2\ lb[/tex]

For this case we have that by definition, the circumference of a circle is given by:

[tex]C = \pi * d[/tex]

Where:

d: It is the diameter of the circle

They tell us that:

[tex]d = \frac {35} {2}\\\pi = \frac {22} {7}[/tex]

Substituting:

[tex]C = \frac {22} {7} * \frac {35} {2}\\C = \frac {770} {14}\\C = 55 \ cm[/tex]

Thus, the circumference of the circle is 55 centimeters.

Answer:

Option B

Answer:

55 im not sure

Step-by-step explanation:

Step-by-step explanation:

h = -490t² + 1470t

When h = 980:

980 = -490t² + 1470t

Simplifying:

0 = -490t² + 1470t - 980

0 = t² - 3t + 2

Factoring:

0 = (t - 1) (t - 2)

t = 1, 2

There are two solutions because the rocket first reaches the height of 980 cm as it's going up at 1 second, then it reaches that height again as it's coming down at 2 seconds.

The rocket reaches a height of 980 cm at 1 second and 2 seconds. The equation is factored to find these solutions, symbolizing the ascent and descent of the rocket.

To find when the height of the rocket is 980 cm, we set the height formula equal to 980 cm:

h = -490t² + 1470t = 980.

We then solve the equation by factoring:

-490t² + 1470t - 980 = 0
Divide by -10:
t²- 3t + 2 = 0
Factor:
(t - 1)(t - 2) = 0

Setting each factor to zero gives us the solutions:

t = 1 second

t = 2 seconds

There are two solutions because the rocket reaches 980 cm twice: once on its way up and once on its way down. This is demonstrated by the positive and negative components of the parabolic trajectory represented by the quadratic equation.

Answer:

3 3/4

Step-by-step explanation:

I've no idea if this is right, but this is how I'd do it:

For 2 batches you need 2 1/2 cups of blueberry

So, half of that is 1 and 1/4

So the total is 3 3/4

Answer:

The answer is 3 3/4

Step-by-step explanation:

I got it right on my test.

Answer:

There are a total of 4 + r + 6 marbles in the bag

The probability of a blue is  [tex]\frac{4}{10+r}[/tex]  

The probability of a red is  [tex]\frac{r}{10+r}[/tex]  

The probability of choosing a blue, replacing it and then a red is

[tex]\frac{4}{10+r}[/tex] × [tex]\frac{r}{10+r}[/tex]  =  [tex]\frac{4r}{100+20r+ r^{2} /[tex]

Step-by-step explanation:

Answer:

Step-by-step explanation:

5

The average yield per plant is 4.00 pints, and the standard deviation is approximately 0.35 pints.

To calculate the average yield per plant:

1. Add up all the yields: 4 + 3.5 + 4.5 + 4.2 + 3.8 = 19.0 pints.

2. Divide the total yield by the number of plants (5): 19.0 / 5 = 3.8 pints per plant.

To calculate the standard deviation:

1. Find the mean (average) yield per plant, which we already calculated as 3.8 pints.

2. Calculate the squared differences from the mean for each yield:

  - Plant 1: [tex](4 - 3.8)^2[/tex] = 0.04

  - Plant 2:[tex](3.5 - 3.8)^2[/tex] = 0.09

  - Plant 3: [tex](4.5 - 3.8)^2[/tex] = 0.49

  - Plant 4:[tex](4.2 - 3.8)^2[/tex] = 0.16

  - Plant 5:[tex](3.8 - 3.8)^2[/tex] = 0

3. Calculate the average of these squared differences: (0.04 + 0.09 + 0.49 + 0.16 + 0) / 5 = 0.156.

4. Take the square root of the variance to get the standard deviation: √0.156 ≈ 0.35 pints.

The average yield per plant provides a measure of the central tendency of the data set, indicating the typical yield for one plant.

The standard deviation measures the spread or variability of the data points around the mean. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation indicates more variability. In this case, the standard deviation of approximately 0.35 pints suggests that the yields are relatively close to the average of 3.8 pints per plant, with minimal variability.

Complete question

A homeowner has 5 cherry tomato plants in her garden. Over the course of the season the yields (in pints of tomatoes per plant) are: Plant 1 2 3 4 5 Yield 4 3.5 4.5 4.2 3.8 What is the average yield per plant, and what is the standard deviation (to two decimal places)?

Answer:

The box of 32oz is cheaper by 2 cents per ounce.

Step-by-step explanation:

1. 3.84/24 ($0.16/oz)

2. 4.48/32 ($0.14/oz)

___

Hope this helps you! :)

Using technology, the regression equation is

y=9.08292(1.09965)ˣ. 

The number of recommendations after 60 days is 2113.277 ≈ 2113.

Explanation

I used Desmos to graph the points, then entered the expression y₁~a*bˣ to calculate the regression.

I then substituted 60 in for x in the regression equation to get the answer of 2113.

Answer:

D. 158

Step-by-step explanation:

Fill in the x's with a 15:

[tex]y=-2(15)^2+40(15)+8[/tex]

so y = 158

The best prediction for the number of new cases in year 15 is 158.

Any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power solve for x in the quadratic equation.

Here, Given quadratic equation

                 y = -2x² + 40x + 8

 at put the value of x = 15, we get

               y(15) = -2 X (15)² + 40 X 15 + 8

               y(15) = -2 X 225 + 600 + 8

               y(15) = -450 + 600 + 8

              y(15) = 150 + 8

              y(15) = 158

Thus, the best prediction for the number of new cases in year 15 is 158.

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Answer:

D. (–3, –6) and (5, 10)

Step-by-step explanation:

The system has equations:

[tex]y=2x[/tex]...(1)

and

[tex]y=x^2-15[/tex]...(2)

Equate both equations;

[tex]x^2-15=2x[/tex]

Rewrite in standard form:

[tex]x^2-2x-15=0[/tex]

We factor to obtain:

[tex](x+3)(x-5)=0[/tex]

By the zero product principle;

[tex](x+3)=0,(x-5)=0[/tex]

[tex]\implies x=-3,x=5[/tex]

When x=-3, y=2(-3)=-6

This yields the ordered pair (-3,-6).

When x=5, y=2(5)=10

This yields the ordered pair (5,10).

The correct choice is D.

Answer:

Correct: A, D, F

Step-by-step explanation:

Consider right triangle EFG.

By the definition of the cosine function,

[tex]\cos \angle E=\dfrac{EG}{EF}\\ \\\cos 30^{\circ}=\dfrac{EG}{EF}\\ \\\dfrac{\sqrt{3}}{2}=\dfrac{EG}{EF}\\ \\EG=\dfrac{\sqrt{3}}{2}EF[/tex]

Option F is correct.

By the definition of the sine function,

[tex]\sin \angle E=\dfrac{FG}{EF}\\ \\\sin 30^{\circ}=\dfrac{GF}{EF}\\ \\\dfrac{1}{2}=\dfrac{GF}{EF}\\ \\GF=\dfrac{1}{2}EF\\ \\EF=2FG[/tex]

Option D is correct.

By the definition of tangence,

[tex]\tan \angle E=\dfrac{FG}{EG}\\ \\\tan 30^{\circ}=\dfrac{GF}{EGF}\\ \\\dfrac{1}{\sqrt{3}}=\dfrac{GF}{EG}\\ \\GF=\dfrac{1}{\sqrt{3}}EGF\\ \\EG=\sqrt{3}FG[/tex]

Option A is correct

Your answer is A) Acute

Answer:

Read the explanation

Step-by-step explanation:

- The Fibonacci Sequence is the series of numbers:

 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

- The next number is found by adding up the two numbers before it

# Example:

- The 2 is found by adding the two numbers before it (1+1)

- The 3 is found by adding the two numbers before it (1+2),

- The 5 is (2+3) and so on

- The rule of it is x(n) = x(n-1) + x(n-2) where

# x(n) is term number ⇒ n

# x(n-1) is the previous term ⇒ (n-1)

# x(n-2) is the term before that ⇒ (n-2)

- Example: the 8th term is  the 7th term plus the 6th term:

 x(8) = x(7) + x(6)

# Note: When we take any two consecutive Fibonacci Numbers,

 their ratio is very close to the Golden Ratio φ

- The Golden Ratio φ is approximately 1.618034...

- We can calculate any Fibonacci Number using the Golden Ratio:

  x(n) = [φ^n - (1 - φ)^n]/√5

- The answer is a whole number, exactly equal to the addition of the

  previous two terms.

# There is an interesting patterns in Fibonacci sequence:

  every nth number is a multiple of x(n)

- Example:

* x3 = 2 ⇒ every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)

* x4 = 3 ⇒ Every 4th number is a multiple of 3 (3, 21, 144, ...)

* x5 = 5 ⇒ Every 5th number is a multiple of 5 (5, 55, 610, ...)

- The man who invented it.

 His real name was Leonardo Pisano Bogollo, and he lived between

 1170 and 1250 in Italy.

- Fibonacci was his nickname

- Fibonacci sequence is it used for:

# Reflects patterns of growth spirals (a spiral curve , shape , pattern ,

  object) found in nature

# It is the closest approximation in integers to the logarithmic spiral

  series, which follows the same rule as the Fibonacci sequence

The Fibonacci sequence is a series of numbers important in mathematics and nature, introduced to European mathematics by Leonardo of Pisa, also known as Fibonacci. It describes various natural phenomena, appears in plants' growth patterns, contributes to arts and sciences, and has practical uses in algorithms and market analysis.

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth. This mathematical concept was brought to European mathematics by Leonardo of Pisa, known as Fibonacci. He described this sequence in his book 'Liber Abaci' in 1202, although the sequence had been previously described in Indian mathematics.

The Fibonacci sequence is not just a number theory curiosity but has many applications and appears in the natural world. Notably, it is used to describe various phenomena in quantum computing, stock market analysis, and in nature, such as the branching of trees, the arrangement of leaves on a stem, or the fruitlets of a pineapple. In the context of spirals and botany, the mechanism of the spirals of plants follows the Fibonacci sequence, as explored in resources like 'Doodling in Math: Spirals, Fibonacci, and Being a Plant' by Vihart and 'How to Count the Spirals' by MoMath: National Museum of Mathematics.

The Fibonacci sequence also finds itself being applied to areas such as computer algorithms, especially in recursive algorithms where an operation needs to be repeated on a smaller scale. It also helps build a bridge between mathematics and the arts, showing how mathematics can inspire beautiful patterns and structures in man-made and natural art forms. The prevalence of the Fibonacci sequence and the golden ratio in aesthetic compositions demonstrates the union between form and number that fascinates both mathematicians and artists alike.

Answer:

Step-by-step explanation:

A parameter describes a population, and a statistic describes a sample.

The first one is a statistic.  A sample was surveyed.

The second one is a parameter.  It describes the entire soccer team.

The third one is a statistic.  A sample was surveyed.

The fourth one is a parameter.  It describes the entire golf team.

Answer:

(x + 5)² + (y + 8)² = 49

Step-by-step explanation:

The equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k) are the coordinates of the centre and r is the radius

here (h, k) = (- 5, - 8), so

(x - (-5))² + (y - (- 8))² = r², that is

(x + 5)² + (y + 8)² = r²

The radius is the distance from the centre to a point on the circle

Use the distance formula to calculate r

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (- 5, - 8) and (x₂, y₂ ) = (2, - 8)

r = [tex]\sqrt{(2+5)^2+(-8+8)^2}[/tex] = [tex]\sqrt{7^2+0^2}[/tex] = [tex]\sqrt{49}[/tex] = 7

r = 7 ⇒ r² = 7² = 49

Hence

(x +5)² + (y + 8)² = 49 ← equation of circle