Problem 20 - Precalculus: A Concise Course

Question

The sun is \(20^{\circ}\) above the horizon. Find the length of a shadow cast by a park statue that is 12 feet tall.

Step-by-Step Solution

Answer

The length of the shadow cast by the statue is approximately 33.07 feet.

Step 1: Identify the known variables

The height of the statue is 12 feet which is the length of the side of the triangle opposite to the given angle. The angle of elevation of the sun is \(20^{\circ}\) which is the angle at which the statue 'sees' the sun or, equivalently, the shadow 'sees' the end point of the statue.

Step 2: Apply the tangent formula

Now, the formula of tangent for a right angle triangle is \( \tan {\theta} = \frac {opposite side} {adjacent side}\). Using this formula, we can state that the length of the shadow (which is the adjacent side to the angle) is equal to the length of the side opposite to the angle (height of the statue) divided by the tangent of the angle of the sun’s elevation.

Step 3: Calculate the length of the shadow

Now substitute the given values into the formula. We have \( \frac {12 feet} {\tan {20^{\circ}}} \). Calculate the value of the tangent of \(20^{\circ}\) and divide 12 by that value. This will give the length of the shadow.