Chapter 4.1, Problem 62E

a.

To determine

To find: On what day is the temperature increasing the fastest?

Answer to Problem 62E

On ${101}^{\text{st}}$ daythe temperature is increasing the fastest.

Explanation of Solution

Given: The equation that approximates the temperature on the day $x$ is,

  $y=37\sin \left[\frac{2\pi }{365}\left(x-101\right)\right]+25$

  $\begin{array}{c}y=37\sin \left[\frac{2\pi }{365}\left(x-101\right)\right]+25\\ \frac{dy}{dx}=\frac{d}{dx}\left(37\sin \left[\frac{2\pi }{365}\left(x-101\right)\right]+25\right)\\ \frac{dy}{dx}=\frac{74\pi }{365}\cos \left(\frac{2\pi }{365}\left(x-101\right)\right)\\ \frac{d^{2}y}{d{x}^2}=-\frac{148{\pi }^2}{133225}\sin \left(\frac{2\pi }{365}\left(x-101\right)\right)\\ \frac{d^{2}y}{d{x}^2}=0\end{array}$

So we can say that $-\frac{148{\pi }^2}{133225}\sin \left(\frac{2\pi }{365}\left(x-101\right)\right)=0$ ,

By solving for $x$ we get $x=101$ .

On ${101}^{\text{st}}$ daythe temperature is increasing the fastest.

b.

To determine

To find: How many degrees per day is the temperature increasing when it is increasing at its fastest?

Answer to Problem 62E

When the temperature is increasing at its fastest is $\frac{74\pi }{365}\text{ degrees per day}$ .

Explanation of Solution

Given: The equation that approximates the temperature on the day $x$ is,

  $y=37\sin \left[\frac{2\pi }{365}\left(x-101\right)\right]+25$

  $\begin{array}{l}\frac{dy}{dx}=\frac{74\pi }{365}\cos \left(\frac{2\pi }{365}\left(101-101\right)\right)\\ \frac{dy}{dx}=\frac{74\pi }{365}\end{array}$

When the temperature is increasing at its fastest is $\frac{74\pi }{365}\text{ degrees per day}$ .