Question

The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth.

Short answer

The length of the flag pole’s shadow is approximately 6.3 feet and the height of the flag pole is 18.9 feet.

Step-by-step solution

Step 1. Draw a figure by using the given information.

The height of the flag pole is three times the length of its shadow.

Let x be the length of the shadow. Then the height of the flag pole is 3x.

The distance between the end of the shadow and the top of the flag pole is 20 feet. The required figure is:

Step 2. Form a quadratic equation.

According to the Pythagoras Theorem:

$(\text{Base}{)}^2+(\text{Perpendicular}{)}^2=(\text{Hypotenuse}{)}^2$

Use the Pythagoras Theorem to write the equation.

$x^2+(3x{)}^2=(20{)}^2$

Step 3. Solve the equation.

The above equation can be written as:

$\begin{array}{rcl}{x}^2+9x^2& =& 400\\ 10x^2& =& 400\\ x^2& =& \frac{400}{10}\\ x& =& \pm \sqrt{40}\\ x& \approx & 6.3,-6.3\end{array}$

Lenght and height cannot be a negative value. So, $x\approx 6.3$. The height of the flag pole is:

$\begin{array}{rcl}3x& =& 3(6.3)\\ & =& 18.9\end{array}$

Step 4. Conclusion.

The length of the flag pole’s shadow is approximately 6.3 feet and the height of the flag pole is 18.9 feet.