Question

suppose the sun is shining on a pole that is 5 feet tall. if the pole casts a shadow that is x feet long, the angle of elevation of the sun is given by: $\theta=arctanleft(\frac{5}{x}
ight)$. what is the rate of change in the angle of elevation with respect to the length of the shadow at the moment when the shadow is 7 feet long? radians per foot

Explanation:

Step1: Differentiate $\theta$ with respect to $x$

We know that $\theta=\arctan(\frac{5}{x})$. Using the chain - rule and the derivative formula for $y = \arctan(u)$, where $y^\prime=\frac{u^\prime}{1 + u^{2}}$. Let $u=\frac{5}{x}=5x^{-1}$, then $u^\prime=- 5x^{-2}=-\frac{5}{x^{2}}$. So $\frac{d\theta}{dx}=\frac{-\frac{5}{x^{2}}}{1 + (\frac{5}{x})^{2}}$.

Step2: Simplify the derivative

First, expand the denominator: $1+(\frac{5}{x})^{2}=1+\frac{25}{x^{2}}=\frac{x^{2}+25}{x^{2}}$. Then $\frac{d\theta}{dx}=\frac{-\frac{5}{x^{2}}}{\frac{x^{2}+25}{x^{2}}}=-\frac{5}{x^{2}+25}$.

Step3: Evaluate at $x = 7$

Substitute $x = 7$ into $\frac{d\theta}{dx}$. We get $\frac{d\theta}{dx}\big|_{x = 7}=-\frac{5}{7^{2}+25}=-\frac{5}{49 + 25}=-\frac{5}{74}$ radians per foot.

Answer:

$-\frac{5}{74}$ radians per foot