Chapter 12, Problem 25PT

To determine

To find: The equation for the height of point $h$ as a function of time $t$ .

Answer to Problem 25PT

  $h=10\cos \frac{2\pi }{45}t+10$

Explanation of Solution

Given information:

Diameter of water wheel $=20$ feet

It makes one revolution in $45$ seconds.

Height of the top of the wheel represents the height at time $0$ .

Given

Height of the top of the wheel represents the height at time $0$ .

So, we need to use Cosine function as at time $0$ the point is at the heights place.

The height of water wheel is between $0$ and $20$

So, its vertical shift is $\frac{0+20}{2}=\frac{20}{2}=10$

We know

Amplitude is the difference between maximum height and Vertical shift

We choose cosine function and our maximum is at the right place,

So, there will not be phase shift.

Given

It makes one revolution in $45$ seconds.

So, period is $45$ .

We know

Period $=\frac{2\pi }{b}$

  $\begin{array}{l}\Rightarrow 45=\frac{2\pi }{b}\\ \Rightarrow b=\frac{2\pi }{45}\end{array}$

Therefore,

We know

If equation in the form

  $y=a\cos \left[b\left(x+h\right)\right]+k$

Then

Amplitude $=\left|a\right|$

Period $=\frac{2\pi }{\left|b\right|}$

Phase shift $=-h$

Vertical shift $=k$

On substituting values

We get

  $h=10\cos \frac{2\pi }{45}t+10$

Graph of above equation is

  

Therefore,

Required equation is

  $h=10\cos \frac{2\pi }{45}t+10$