Chapter 5.2, Problem 62E

Shadow Length

The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is $\theta$ can be modeled by the equation

$s=\frac{h\sin \left({90}^{\circ }-\theta \right)}{\sin \theta },0^{\circ }<\theta \le {90}^{\circ }$.

(a) Verify that the expression for s is equal to h $\cot \theta$.

(b) Use a graphing utility to create a table of the lengths s far different values of $\theta$. Let h = 5 feet. (c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow.

(d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is ${90}^{\circ }$?