It is afternoon. How fast is the length of the shadow of a 72 ft tall tree changing when \( \theta \), the angle of elevation of the sun, is \(\frac {\pi }{6}\)? Assume that the rate at which the angle of elevation is decreasing is 0.25 rad/hr.

STEP 1

Introduce a variable \(x\) that denotes the length of the shadow of the tree.Then sketch and label the figure.

STEP 2

Complete the following statements.

The known rate is

\[ \dd t \theta =\answer { - 0.25} \text { rad}/hr \]

The rate to be determined is

\[ \dd t \answer {x} \text { , when } \theta =\frac {\pi }{6} \]

STEP 3

Write an equation that relates all relevant variables.

\[ \tan (\theta )=\answer {72/x} \]

STEP 4

Differentiate both sides of the equation above with respect to \(t\) (differentiate the left side first) .

\[ \answer {\sec ^{2}{(\theta )}}\cdot \dd t \answer {\theta } =\answer {-72/x^{2}} \cdot \dd t \answer {x} \]

STEP 5

Evaluate. Now, in order to evaluate the equation at the instant when \(\theta =\frac {\pi }{6}\), we have to find the value of \(x\) at that instant!

\[ \left [x\right ]_{\theta =\frac {\pi }{6}}=\frac {\answer {72}}{\tan {\left (\frac {\pi }{6}\right )}} \]
\[ \answer {\sec ^{2}{\left (\frac {\pi }{6}\right )}}\cdot \answer {-0.25} =\frac {-\tan ^2{\left (\frac {\pi }{6}\right )}}{\answer {72}}\cdot \left [\dd t {x} \right ]_{\theta =\frac {\pi }{6}} \]

STEP 6

Solve for \( \left [\dd t {x} \right ]_{\theta =\frac {\pi }{6}}\).

\[ \left [\dd t {x} \right ]_{\theta =\frac {\pi }{6}}=\answer {72}\text { ft/hr}. \]