Chapter 13.8, Problem 36PPS

To determine

Find the equation for the height of a car h as a function of time t ,minutes and also draw the graph.

Answer to Problem 36PPS

The equation of the car is $h=520\cos \frac{\pi }{15}\theta +5$ .

Explanation of Solution

Given: It is given in the question that a ferris wheel has a diameter of approximately $520ft$ and makes one complete revolution in 30 minutes. Suppose the lowest car on the Ferris wheel is $5$ feet from the ground. Let the height at the top of the wheel represent the height at time zero.

Concept Used:

In this, use the concept of the general equation for cosine $y=a\cos b\theta +d$ , where a is amplitude, $\frac{2\pi }{\left|b\right|}$ is period and d is the vertical shift.

Calculation: Let the equation be $h=a\cos b\theta +d$ .

Here, the vertical shift is $5$ feet because the car height is given.

  $d=5$

Since, the amplitude is same as the diameter of the wheel because amplitude is also means the full distance, so $a=520$ .

Since the period is $30\min$ , so

  $\begin{array}{l}\frac{2\pi }{b}=30\\ b=\frac{2\pi }{30}\\ b=\frac{\pi }{15}\end{array}$

Now, put all the values in the equation,

  $h=520\cos \frac{\pi }{15}\theta +5$

The graph of the function is :

  

Conclusion:

The equation is $h=520\cos \frac{\pi }{15}\theta +5$ .