Chapter 13.8, Problem 41PPS

a.

To determine

Find the equation to represent the height of the horse h as a function of time t seconds.

Answer to Problem 41PPS

The equation is $y=9\sin (\frac{2\pi }{7})t+46$ .

Explanation of Solution

Given: It is given in the question that a horse on a carousel goes up and down three times as the carousel makes one complete rotation. The maximum height of the horse is $55inches$ , and the minimum height is $37inches$ . The carousel rotates once every $21\sec$ Assume that the horse starts and stops at its median height.

Concept Used:

In this, use the concept of general equation of sine $y=a\sin b(t)+h$ , where a is the amplitude, $\frac{2\pi }{\left|b\right|}$ is the period and h is the midline and it is the average between the maximum and minimum value, t is the time.

Calculation: Let the equation be $y=a\sin b(t)+h$ .

First find the midline h,

  $\begin{array}{l}\frac{55+37}{2}=\frac{92}{2}=46\\ h=46\end{array}$

Since, the amplitude is the difference between midline and maximum and minimum value.

  $\begin{array}{l}55-46=9\\ a=9\end{array}$

It takes $7$ seconds because $3$ periods occur during $21$ seconds $(\frac{21}{7}=3)$ .

So, the period is $7$ .

Since,

  $\begin{array}{l}\frac{2\pi }{\left|b\right|}=7\\ b=\frac{2\pi }{7}\end{array}$

Put the above value in the equation $y=a\sin b(t)+h$

The equation is $y=9\sin (\frac{2\pi }{7})t+46$ .

Conclusion:

The function is $y=9\sin (\frac{2\pi }{7})t+46$ .

b.

To determine

Draw the graph of the function.

Explanation of Solution

Given:

It is given in the question that the function is $y=9\sin (\frac{2\pi }{7})t+46$ .

Graph:

  

Interpretation:

Here, put the function $y=9\sin (\frac{2\pi }{7})t+46$ in the graphing calculator, it gives the required graph in the right way. It gives sine curve.

c.

To determine

Calculate height of the horse after eight seconds.

Answer to Problem 41PPS

The height of the horse is $53inch$ .

Explanation of Solution

Given:

It is given in the question that the function is $y=9\sin (\frac{2\pi }{7})t+46$ .

Concept Used:

In this, use the concept of substituting the given value in the function and get the solution.

Calculation: Here, the function $y=9\sin (\frac{2\pi }{7})t+46$ .

Substitute the value $t=8$ in the above function,

  $\begin{array}{l}y=a\sin b(t)+h\\ y=9\sin (\frac{2\pi }{7}t)+46\\ y=9\sin (\frac{2\pi }{7}\cdot 8)+46\\ y=9\sin (2.28\pi )+46\\ y=9\cdot 0.78+46\\ y=7.02+46\\ y\approx 53inch\end{array}$

Conclusion:

The height of the horse is $53inch$ .