The volume of a rectangular prism is 960 cubic inches. If the dimensions of the base are doubled and the height remains the same to create a new prism, what will be the volume of the new rectangular prism in cubic inches?

Answer:

 GENERAL EXPLICIT SEQUENCE IS GIVEN   [tex]a_n = (14.74)(r)^{n-1)}[/tex]

Step-by-step explanation:

Let n be the number of year the data is recorded in.

a: The number of raccoons taken initially.

r: The multiplying factor

[tex]a_n[/tex]  : The number of raccoon in the nth year.

As given:  [tex]a_6 = 45, a_8 = 71[/tex]

Now, as the given situation can be expressed as GEOMETRIC SERIES:

[tex]a_n = a r^{(n-1)}[/tex]

Applying the same to given terms, we get:

[tex]a_6 = a r^{(6-1)} = ar^5 = 45\\\implies ar^5 = 45[/tex]

[tex]a_8 = a r^{(8-1)} = ar^7 =71\\\implies ar^7 = 71[/tex]

Dividing both equations, we get:

[tex]\frac{ar^7}{ar^5} = \frac{71}{45} \\\implies r^2 = 1.58\\\implies r = 1.25[/tex]

So, the first term [tex]a = \frac{45}{(1.25)^5} = 14 .74 \approx 15[/tex]

So, the GENERAL EXPLICIT SEQUENCE IS GIVEN as:  [tex]a_n = (14.74)(r)^{n-1)}[/tex]

Answer:

Deepak: 4 t-shirts,

Tarun: 12 t-shirts,

Mahesh: 14 t-shirts.

Step-by-step explanation:

Let x represent number of t-shirts that Deepak has.

Please consider the complete question.

Tarun has 4 t-shirt more than twice the number of T-shirts Deepak has. Mahesh has 2 more than thrice the number of T-shirts Deepak has. If the ratio of the t-shirt that Tarun and Mahesh have is 6 : 7, find out the number of the t-shirt each of them has.​

Since Tarun has 4 t-shirt more than twice the number of T-shirts Deepak has, so the number of t-shirts that Tarun has would be [tex]2x+4[/tex].

We are also told that Mahesh has 2 more than thrice the number of T-shirts Deepak has. So the number of t-shirts that Mahesh has would be [tex]3x+2[/tex].

Since the ratio of the t-shirt that Tarun and Mahesh have is 6 : 7, so we can represent this information in an equation as:

[tex]\frac{2x+4}{3x+2}=\frac{6}{7}[/tex]  

Cross multiply:

[tex]6(3x+2)=7(2x+4)[/tex]

[tex]18x+12=14x+28[/tex]

[tex]18x-14x+12-12=14x-14x+28-12[/tex]

[tex]4x=16[/tex]

[tex]x=\frac{16}{4}=4[/tex]

Therefore, Deepak has 4 t-shirts.

The number of t-shirts that Tarun has would be [tex]2x+4\Rightarrow 2(4)+4=4+4=12[/tex]

Therefore, Tarun has 12 t-shirts.

The number of t-shirts that Mahesh has would be [tex]3x+2\Rightarrow 3(4)+2=12+2=14[/tex]

Therefore, Mahesh has 14 t-shirts.

Answer:

The cheapest route to visit each city and return home again to Athens is:

A→B→C→D→A  or A→D→C→B→A.

Step-by-step explanation:

The Algorithm of Brute Force

Let Athens ⇒A , Buford ⇒B , Cuming ⇒ C , Dacula ⇒ D

There are 6 routes to visit each city and return home again to Athens.

Route 1: A→B→C→D→A = 70 + 25 + 30 + 60 = $185

Route 2: A→B→D→C→A = 70 + 70 + 30 + 50 = $220

Route 3: A→C→B→D→A = 50 + 25 + 70 + 60 = $205

Route 4: A→C→D→B→A = 50 + 30 + 70 + 70 = $220

Route 5: A→D→B→C→A = 60 + 70 + 25 + 50 = $205

Route 6: A→D→C→B→A = 60 + 30 + 25 + 70 = $185

By checking the previous routes:

The cheapest charge will be $185 and it will be for the route

A→B→C→D→A  or A→D→C→B→A.

Answer: the swimming pool can hold 31707.72 gallons of water.

Step-by-step explanation:

The swimming pool is shaped like a cylinder. The formula for determining the volume of a cylinder is expressed as

Volume = πr²h

Where

r represents the radius of the cylindrical swimming pool.

h represents the height of the pool.

π is a constant whose value is 3.14

Therefore, volume of the swimming pool is

Volume = 3.14 × 15² × 6 = 4239 cubic feet

if one cubic foot holds 7.48 gallons of the water, then the number of gallons of water that the swimming pool can hold is

4239 × 7.48 = 31707.72 gallons of water

Answer:

the swimming pool can hold 31707.72 gallons of water.

Step-by-step explanation:

Answer:

59/36π

Step-by-step explanation:

We know that an angle is measured in either degrees or radians and The arc's angle measurement, taken at the center of the circle the arc is part of, is measured in degrees (or radians)

Let's convert 90 degrees into radians

295° = 295 * π/180 = 59/36π

Answer:

1220

Step-by-step explanation:

Given that in a game, a player earns 100 points for each question answered correctly and earns −30 points for each question answered incorrectly.

Let x be the no of questions correctly answered .  Then 20-x would be the question wrongly answered since total number of questions = 14+6 =20

Points gained for correct answer = 100(x) = 100x

Points lost for wrong answer = -30(20-x) = -600+30x

So total points gained when x questions are answered right

= [tex]100x-600+30x\\= 130x-600[/tex]

A player answered 14 questions correctly and 6 questions incorrectly.

Here x =14

Hence we substitute x =14 to get total points earned

Total points earned

[tex]= 130(14)-600\\= 1820-600\\=1220[/tex]

Answer:

The 3rd option Summation(4^i-4)

Step-by-step explanation:

Summation(4^i-4)

When i = 1

4^I-4 = 4^1-4 = 4^-3 = 1/64

When i = 2

4^i-4 = 4^2-4 = 4^-2 = 1/16

When i = 3

4^i-4 = 4^3-4 = 4^-1 = 1/4

When i = 4

4^i-4 = 4^4-4 = 4^0 = 1

When i = 5

4^i-4 = 4^5-4 = 4^1 = 4

Answer:

[tex]\sum _{i=1}^54^{i-4}[/tex]

Step-by-step explanation:

[tex]a_{1} \\[/tex] = First term

In this case, our first term is [tex]\frac{1}{64}[/tex]

The ratio of all of the adjacent terms is 4

[tex]a_{1} \\[/tex] = [tex]4^1^-^4[/tex]

[tex]a_{1}=\frac{1}{64}[/tex]

~Hope this helps!~

The expression d^2 - 4d + 4 factors to (d - 2)^2.

To factor the expression d^2 + 4d + 4, we are looking for two binomials that will multiply together to give us the original quadratic expression. These binomials will be of the form (d - a)^2 because the last term is a perfect square (4 = 22) and the middle term is twice the product of the square roots of the first and last terms.
Here's the step-by-step factoring:
1. Identify the square root of the first term, which is d.
2. Identify the square root of the last term, which is 2.
3. Since our middle term is negative, we use negative signs in our binomials.
4. The factored form is (d - 2)^2, as this will expand to d^2 - 2*d*2 + 22, which simplifies to d^2 - 4d + 4.

Answer:

Answer is; Evaluating

Step-by-step explanation:

Student self-assessment involves students evaluating their own work and learning progress.

Through self-assessment evaluation, students can:

* See where their knowledge is weak

* See where to focus their attention in learning.

* Set realistic goals

* Revise their work

* Track their own progress.

In Rachel's case, after writing her midterm exams, she already set her goal grade. So when she got her midterm grades, she needed to evaluate her work to find out her performance.

Therefore, according to the question, Rachel is EVALUATING her performance.

Answer:

Diameter

Step-by-step explanation:

Knowing that a diameter is the largest chord as it passes through the center point of a circle, if the definition of the chord is a line segment with it's two endpoints on the circle (here by circle we mean circumference) then a diameter is a line segment that passes through the center and has endpoints on the circumference.

Answer:

Therefore the critical value= 2.073

Step-by-step explanation:

The number of observation = 23.

For the mean of the population the confidence interval  = 95%.

Here mean and stander deviation of the distribution is not given. So we use t distribution.

T distribution is called as student's t distribution.

Sample number =n =23.

Confidence level = c= 95% =0.95

The degree of freedom is sample size decreased by 1

df=n-1 = 23-1 =22.

The critical value [tex]t^*[/tex] can be found in the row df = 22 and column with 0.95 of the T distribution table .

[tex]t^*[/tex] = 2.073

Therefore the critical value= 2.073

To calculate the critical value for a 95% confidence interval when the sample size is small (less than 30), we need to use the t-distribution. The critical value from the t-distribution will depend on the sample size (specifically, the degrees of freedom) and the desired level of confidence.
Here's how you could calculate this step by step without using the Python function mentioned:
**Step 1: Identify the desired confidence level.**
For a 95% confidence interval, we are interested in capturing the central 95% of the t-distribution.
**Step 2: Determine the degrees of freedom.**
The degrees of freedom (df) for a t-distribution is equal to the sample size minus 1. So with 23 observations, df = 23 - 1 = 22.
**Step 3: Find the critical t-value.**
We want to find the critical t-value for the t-distribution with 22 degrees of freedom that corresponds to the 95% confidence interval. This means we want to find the t-value such that 95% of the distribution lies between -t and +t. Because the t-distribution is symmetric, we can look up the critical value for 97.5% (to split the remaining 5% evenly on both tails of the distribution).
Using a t-distribution table (often found in the appendices of statistics textbooks) or a statistical computing resource, you would find the t-value that corresponds to a cumulative probability of 0.975 with 22 degrees of freedom.
**Step 4: Interpret the table or resource correctly.**
If you were looking at a table, you would look down the degrees of freedom column until you find 22, then right to the column that represents the 97.5% cumulative probability (remember, this is for the two-tailed test). That entry is the critical t-value that corresponds to a 95% confidence interval.
**Step 5: Use the critical t-value for constructing the interval.**
Once you have the critical t-value, you would use it to construct the confidence interval for the population mean by multiplying this t-value by the standard error of the sample mean and then add and subtract this value from the sample mean.
**Important Note:**
Please be aware that the exact t-value varies depending on the source of the statistical tables or the statistical software being used. The value also depends on the precision (number of decimals) presented in the table.
If you perform these steps with a standard statistical table or software, you should find that the critical t-value for a 95% confidence interval with 22 degrees of freedom is approximately 2.074.

Option A: [tex]\frac{10}{9}[/tex] is the solution of x

Explanation:

The given expression is [tex]\frac{7}{(x+2)}+\frac{11}{(x-5)}=\frac{7}{(x+2)(x-5)}[/tex]

We need to determine the value of x.

The value of x can be determined by solving the expression for x.

Taking LCM , we get,

[tex]\frac{7(x-5)+11(x+2)}{(x+2)(x-5)}=\frac{7}{(x+2)(x-5)}[/tex]

Since, the denominator is common for both sides of the equation, let us cancel the denominator.

Thus, we have,

[tex]7(x-5)+11(x+2)=7[/tex]

Multiplying the terms within the bracket, we get,

[tex]7x-35+11x+22=7[/tex]

Adding the like terms, we get,

[tex]18x-13=7[/tex]

Adding both sides of the equation by 13, we have,

[tex]18x=20[/tex]

Dividing both sides of the equation by 18,

[tex]x=\frac{20}{18}[/tex]

Simplifying, we get,

[tex]x=\frac{10}{9}[/tex]

Thus, the solution is [tex]\frac{10}{9}[/tex]

Therefore, Option A is the correct answer.

Answer:

a

Step-by-step explanation:

Answer:

-40x(x+1)

Step-by-step explanation:

Find [g•h](x) and [h•g] (x)

g(x)=2x

h(x)=-10x-10

[g•h](x) = 2x(-10x-10)= -20x^2-20x = -20x(x+1)

[h•g](x) = (-10x-10)2x= -10x(2x)-10(2x) = -20x^2-20x

[g•h](x) and [h•g] (x)

and means addition

-20x^2-20x + (-20x^2-20x)

-20x^2-20x-20x^2-20x

choose like terms

-20x^2-20x^2-20x-20x

-40x^2-40x

-40x(x+1)

Answer:

is d

Step-by-step explanation:

Answer:

2.13%

Step-by-step explanation:

Inaccurate measurement = 5.5cm

Actual measurement = 5.62cm

Difference= actual - inaccurate

=> 5.62 - 5.5

Therefore, the measurement Jocelyn obtained is off by 0.12cm (difference)

Note: error = difference

% diff = error÷ actual measurement × 100

= 0.12/5.62 × 100

% diff. = 2.13 (to the nearest decimal place)

Each friend will get 1.35 pounds of pecans.

To find this, simply divide 16.2 by 12 to find the weight of pecans everyone gets. Thus making 1.35 pounds of pecans the answer.

I hope this helps!

Answer:

It will travel high enough

Step-by-step explanation:

Find the vertex of the parabola:

x=-b/2a

x=-11/2(-16)

x=-11/-32

x=11/32

Plug x=11/32 into quadratic to get the y-coordinate:

h=-16(11/32)^2+11(11/32)+5.5

h=7.391

Since 7.391>7.3, the volleyball will travel high enough (aka. yes)

To determine if the volleyball clears the net, we calculate the maximum height using the vertex of the parabola from the quadratic equation representing the ball's trajectory. By finding the time at the vertex and substituting it back into the equation, we get the maximum height, which needs to be compared with the net's height.

To determine whether the volleyball travels high enough to clear the net, we need to calculate the maximum height reached by the ball using the given quadratic equation h = −16t2 + 11t + 5.5. The maximum height will be at the vertex of the parabola represented by the quadratic function. The t-coordinate of the vertex can be found using the formula t = -b/2a, where a and b are the coefficients from the quadratic term and the linear term respectively.

For the given equation h = -16t2 + 11t + 5.5, a is -16 and b is 11. Thus,

Doing the calculation will reveal the maximum height of the volleyball. If this height is greater than 7.3 feet, the height of the net, then the volleyball clears the net.

https://brainly.com/question/35544673

#SPJ6

Answer:

A.) it costs 140$ for 7 tickets B.) 460$ = 23 tickets

Step-by-step explanation:

If it is 20 dolors for 1 ticket if you buy 7 you do 7 x 20 = 140.

But if you have 460$ then you do the opisit you do 460/20=23

Answer:

Step-by-step explanation:

-(x+2)^2 --> -(x+2)(x+2) --> -(x^2+4x+4) = -x^2-4x-4

Answer:

[tex]x^{2} -4x+4[/tex]

Step-by-step explanation:

[tex]y =-(x+2)^{2}[/tex]

The negative sign multiplies the positive in the bracket

[tex]y=(x-2)^{2}[/tex]

[tex](x-2) X (x-2)[/tex]

[tex]x^{2} -2x-2x+4[/tex]

That gives us

[tex]x^{2} -4x+4[/tex]

Step-by-step explanation:

[tex]-2x^2+10x=-14\qquad\text{divide both sides by (-2)}\\\\\dfrac{-2x^2}{-2}+\dfrac{10x}{-2}=\dfrac{-14}{-2}\\\\x^2-5x=7\qquad(a-b)^2=a^2-2ab+b^2\qquad(*)\\\\x^2-2(x)(2.5)=7\qquad\text{add}\ 2.5^2\ \text{to both sides}\\\\\underbrace{x^2-2(x)(2.5)+2.5^2}_{(*)}=7+2.5^2\\\\(x-2.5)^2=7+6.25\\\\(x-2.5)^2=13.25[/tex]

[tex]\text{If you want the solution, then:}\\\\(x-2.5)^2=13.25\iff x-2.5=\pm\sqrt{13.25}\\\\x-\dfrac{25}{10}=\pm\sqrt{\dfrac{1325}{100}}\\\\x-\dfrac{25}{10}=\pm\dfrac{\sqrt{1325}}{\sqrt{100}}\\\\x-\dfrac{25}{10}=\pm\dfrac{\sqrt{25\cdot53}}{10}\\\\x-\dfrac{25}{10}=\pm\dfrac{\sqrt{25}\cdot\sqrt{53}}{10}\\\\x-\dfrac{25}{10}=\pm\dfrac{5\sqrt{53}}{10}\\\\x-\dfrac{5}{2}=\pm\dfrac{\sqrt{53}}{2}\qquad\text{add}\ \dfrac{5}{2}\ \text{to both sides}\\\\x=\dfrac{5}{2}\pm\dfrac{\sqrt{53}}{2}[/tex]

[tex]\huge\boxed{x=\dfrac{5\pm\sqrt{53}}{2}}[/tex]

Answer: the maximum number of socks she will be able to make from the leftover yarn is 10

Step-by-step explanation:

Let x represent the number of sweaters.

Let y represent the number of socks.

The baby sweaters take 10 feet of yarn and the baby socks take 5 feet of yarn. This means that the total number of feet of yarn needed to make x sweaters and y socks is expressed as

10x + 5y

She has 100 feet of yarn and wants to make 5 sweaters. This means that

10 × 5 + 5y = 100

50 + 5y = 100

5y = 100 - 50 = 50

y = 50/5

y = 10

Answer:

Explanation:

If 112%  interest of bond amount = $27,498,250

∴ 100% principal amount = 27,498,250 X 100/112 = $24,552,008.93

10% bond interest = $24,552,008.93 X 0.10 = $2,455,200.89    

Between October 2018 and August 2019, it amounts = $2,455,200.09 X 11 = $27,007,209.82

The amount accrued up to September 2019 = $(27,007,209.82 + 27,498,250) = $54,505,459.82

From October 2019 to August 2020, it will amount = $(27,007,209.82 + 54,505,459.82) = $81,512669.64

The amount accrued up to September 2020 = $(81,512,669.64 + 27,498,250) = $109,010,919.64

Answer:

The answer to your question is 6.3 units

Step-by-step explanation:

Data

A (-2, 2)

B (4, 4)

Distance = ?

Formula

dAB = [tex]\sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}[/tex]

Substitution

x1 = -2   x2 = 4    y1 = 2   y2 = 4

dAB = [tex]\sqrt{(4 + 2)^{2} + (4 - 2)^{2}}[/tex]

Simplification

dAB = [tex]\sqrt{6^{2} + 2^{2}}[/tex]

dAB = [tex]\sqrt{36 + 4}[/tex]

dAB = [tex]\sqrt{40}[/tex]

Result

dAB = 6.3 units

Answer:

(a) The value of C is 1.

(b) In 2010, the population would be 1.07555 billions.

(c) In 2047, the population would be 1.4 billions.

Step-by-step explanation:

(a) Here, the given function that shows the population(in billions) of the country in year x,

[tex]P(x)=Ca^{x-2000}[/tex]

So, the population in 2000,

[tex]P(2000)=Ca^{2000-2000}[/tex]

[tex]=Ca^{0}[/tex]

[tex]=C[/tex]

According to the question,

[tex]P(2000)=1[/tex]

[tex]\implies C=1[/tex]

(b) Similarly,

The population in 2025,

[tex]P(2025)=Ca^{2025-2000}[/tex]

[tex]=Ca^{25}[/tex]

[tex]=a^{25}[/tex]                    (∵ C = 1)

Again according to the question,

[tex]P(2025)=1.2[/tex]

[tex]a^{25}=1.2[/tex]

Taking ln both sides,

[tex]\ln a^{25}=\ln 1.2[/tex]

[tex]25\ln a = \ln 1.2[/tex]

[tex]\ln a = \frac{\ln 1.2}{25}\approx 0.00729[/tex]

[tex]a=e^{0.00729}=1.00731[/tex]

Thus, the function that shows the population in year x,

[tex]P(x)=(1.00731)^{x-2000}[/tex]     ...... (1)

The population in 2010,

[tex]P(2010)=(1.00731)^{2010-2000}=(1.00731)^{10}=1.07555[/tex]          

Hence, the population in 2010 would be 1.07555 billions.

(c) If population P(x) = 1.4 billion,

Then, from equation (1),

[tex]1.4=(1.00731)^{x-2000}[/tex]

[tex]\ln 1.4=(x-2000)\ln 1.00731[/tex]

[tex]0.33647 = (x-2000)0.00728[/tex]

[tex]0.33647 = 0.00728x-14.56682[/tex]

[tex]0.33647 + 14.56682 = 0.00728x[/tex]

[tex]14.90329 = 0.00728x[/tex]

[tex]\implies x=\frac{14.90329}{0.00728}\approx 2047[/tex]

Therefore, the country's population might reach 1.4 billion in 2047.

Answer:

The answer is 160

Step-by-step explanation:

The value of x is (πr - 20).

The arc length of a circle is the distance between two points on the curve of the circle.

We have,

The measure of an angle formed by two tangents outside the circle.

= Difference of the intercepted arc / 2 ______(1)

Now,

Larger arc = (2πr - x)

Small arc = x

Angle = 20

Substituting in (1).

20 = (2πr - x - x) / 2

40 = 2πr - 2x

20 = πr - x

x = πr - 20

Thus,

The value of x is (πr - 20).

Learn more about arc lengths here:

https://brainly.com/question/16403495

#SPJ7

Answer:

(6,-6)

Step-by-step explanation:

well if you count over to the right 6 and down 6 that points would be

(6,-6)

Answer:

(6,-6)

Step-by-step explanation:

The answer is that because the x-axis for A  is positive and it is 6 points away from the origin. The  y-axis for A should negative this is because the quadrants are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the (x; y) coordinates are I (+; +), II (−; +), III (−; −), and IV (+; −). The point is also 6 units away from the origin.

Answer:

If length of the field is 30 ft, then width is 120 ft.

If the  length of the field is 60 ft, then width is 60 ft.

Step-by-step explanation:

Let us assume the length of the rectangular park = L ft

Let us assume the breadth of the rectangular park = B  ft

Now, AREA of the given park =  L x B

⇒ L x B  = 3,600 sq ft   ... (1)

Also, the perimeter of three sides  = 180 ft

⇒ 2 L +  B  = 180  ..... (2)

Now, from (1) and (2), we get:

L x B  = 3,600

2 L +  B  = 180   ⇒ B  = 180 - 2 L

Substitute this in(1) , we get:

L x B  = 3,600   ⇒ L x (180 - 2 L)  = 3600

[tex]\implies 180 L - 2L^2 = 3600\\\implies L^2 -90L + 1800 = 0\\\implies (L-30)(L-60)= 0[/tex]

⇒ L = 30 or L  = 60

So, if L  = 30  ft , then B = 180 - 2L  =  180 - 60 = 120 ft

So, if L  = 60  ft , then B = 180 - 2L  =  180 - 120 = 60 ft

So, if length of the field is 30 ft, then width is 120 ft.

And if the  length of the field is 60 ft, then width is 60 ft.

Using concepts of the normal distribution, it is found that the statement which is not true is:

C. A​ z-score is an area under the normal curve.

In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Thus, statement C is false, as the p-value is the area under the normal curve, not the z-score.

A similar problem is given at https://brainly.com/question/14243195

Answer:

6

Step-by-step explanation:

f(-4) = g(-10) = f(-10+k)

f(-2) = g(-8) = f(-8+k)

-4 = -10+k

k = -4+10

k = 6

If the line f passes through (-4,0) and (-2,2) it means:

f(-4)=0 and f(-2)=2

If the line g passes through (-10,0) and (-8,2) it means:

g(-10)=0 and g(-8)=2

We can see that

                                                g(-10)=0=f(-4)

                                              g(-10)=0=f(-10+6)

Also,

                                                  g(-8)=2=f(-2)

                                                 g(-8)=2=f(-8+6)

Therefore, k=6

Final answer:

The perimeter of a square with side length (6x-1) is expressed as P = 24x - 4. Substituting x with 3, the perimeter is calculated to be 68 inches.

Explanation:

The expression for the perimeter of a square is given by the formula P=4s, where 's' is the side length of the square. For a square with side length (6x-1) inches, the perimeter would be:

P = 4(6x-1)

This expression can be simplified to:

P = 24x - 4

When x equals 3, we substitute 3 in place of x:

P = 24(3) - 4

P = 72 - 4

P = 68 inches

Therefore, the perimeter of the square when x is 3 is 68 inches.

Answer:

The answers to the question is

(a) Jamie is gaining altitude at 1.676 m/s

(b) Jamie rising most rapidly at t = 15 s

At a rate of 2.094 m/s.

Step-by-step explanation:

(a) The time to make one complete revolution = period T = 15 seconds

Here will be required to develop the periodic motion equation thus

One complete revolution = 2π,

therefore the  we have T = 2π/k = 15

Therefore k = 2π/15

The diameter = radius of the wheel = (diameter of wheel)/2 = 5

also we note that the center of the wheel is 6 m above ground

We write our equation in the form

y = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex]

When Jamie is 9 meters above the ground and rising we have

9 = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex] or 3/5 = [tex]sin(\frac{2*\pi*t}{15} )[/tex] = 0.6

which gives sin⁻¹(0.6) = 0.643 =[tex]\frac{2*\pi*t}{15}[/tex]

from where t = 1.536 s

Therefore Jamie is gaining altitude at

[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] 1.676 m/s.

(b) Jamie is rising most rapidly when   the velocity curve is at the highest point, that is where the slope is zero

Therefore we differentiate the equation for the velocity again to get

[tex]\frac{d^2y}{dx^2} = -5*(\frac{\pi *2}{15} )^2*sin(\frac{2\pi t}{15})[/tex] =0, π, 2π

Therefore [tex]-sin(\frac{2\pi t}{15} )[/tex] = 0 whereby t = 0 or

[tex]\frac{2\pi t}{15}[/tex] = π and t =  7.5 s, at 2·π t = 15 s

Plugging the value of t into the velocity equation we have

[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] - 2/3π m/s which is decreasing

so we try at t = 15 s and we have [tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi *15}{15}) = \frac{2}{3} \pi[/tex]m/s

Hence Jamie is rising most rapidly at t = 15 s

The maximum rate of Jamie's rise is 2/3π m/s or 2.094 m/s.

(a) When Jamie is 9 meters above the ground and rising, she is gaining altitude at approximately 1.68 meters per second. (b) Jamie is rising most rapidly when the cosine function is at its maximum, which happens at the lowest point of the Ferris wheel, and the rate is approximately 2.094 meters per second.

Part (a): Rate at Which Jamie is Gaining Altitude

1. Identify the position function of Jamie on the Ferris wheel:

  The height h of Jamie above the ground as a function of time t can be modeled by the equation of a sinusoidal function:

 [tex]\[ h(t) = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \][/tex]

  Here, 6 meters is the height of the center of the Ferris wheel above the ground, and 5 meters is the radius of the wheel.

2. Differentiate the height function to find the rate of change of height:

  To find the rate at which Jamie is gaining altitude, we need to differentiate h(t) with respect to t:

[tex]\[ h'(t) = \frac{d}{dt} \left( 6 + 5\sin\left(\frac{2\pi}{15}t\right) \right) = 5 \cdot \frac{2\pi}{15} \cos\left(\frac{2\pi}{15}t\right) \][/tex]

  Simplifying,

 [tex]\[ h'(t) = \frac{2\pi}{3} \cos\left(\frac{2\pi}{15}t\right) \][/tex]

3. Determine t when Jamie is at 9 meters above the ground and rising:

[tex]\[ 9 = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \] Solving for \( \sin \left(\frac{2\pi}{15}t\right) \): \[ 3 = 5\sin\left(\frac{2\pi}{15}t\right) \] \[ \sin\left(\frac{2\pi}{15}t\right) = \frac{3}{5} \][/tex]

  Jamie is rising when [tex]\( \cos \left(\frac{2\pi}{15}t\right) > 0 \)[/tex].

4. Find the rate at which Jamie is gaining altitude at this instant:

  Substitute [tex]\(\sin \left(\frac{2\pi}{15}t\right) = \frac{3}{5}\)[/tex] into the derivative [tex]\( h'(t) \)[/tex]:

  [tex]\[ \cos \left(\frac{2\pi}{15}t\right) = \sqrt{1 - \sin^2 \left(\frac{2\pi}{15}t\right)} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \][/tex]

  Thus, the rate of change of height:

[tex]\[ h'(t) = \frac{2\pi}{3} \cdot \frac{4}{5} = \frac{8\pi}{15} \approx 1.68 \text{ meters per second} \][/tex]

Part (b): When Jamie is Rising Most Rapidly

1. Identify when Jamie is rising most rapidly:

  Jamie rises most rapidly when [tex]\( \cos \left(\frac{2\pi}{15}t\right) = 1 \)[/tex], which corresponds to the maximum value of the cosine function.

2. Rate of change of height at maximum rise:

[tex]\[ h'(t) = \frac{2\pi}{3} \cdot 1 = \frac{2\pi}{3} \approx 2.094 \text{ meters per second} \][/tex]