Question

you are trying to determine the height of a lamppost by placing a mirror on level ground, and standing so you can see the top of the lamppost through the mirror. the lamppost is 23 feet away from the mirror, and you are standing 7 feet away in the opposite direction. if you are 6 feet tall, how tall is the lamppost? round your answer to the nearest hundredths. feet

Explanation:

Step1: Set up the proportion

Since the triangles formed are similar (by the law of reflection, the angles of incidence and reflection are equal, creating similar right - angled triangles), we can set up the proportion \(\frac{\text{your height}}{\text{your distance from mirror}}=\frac{\text{lamppost height}}{\text{lamppost distance from mirror}}\). Let \(h\) be the height of the lamppost. We know your height \(y = 6\) feet, your distance from the mirror \(d_1=7\) feet, and the lamppost distance from the mirror \(d_2 = 23\) feet. The proportion is \(\frac{6}{7}=\frac{h}{23}\).

Step2: Solve for \(h\)

Cross - multiply the proportion: \(7h=6\times23\). First, calculate \(6\times23 = 138\). Then, solve for \(h\) by dividing both sides of the equation \(7h = 138\) by 7. So, \(h=\frac{138}{7}\).

Step3: Calculate the value

Using a calculator, \(\frac{138}{7}\approx19.714286\). Rounding to the nearest hundredth (two decimal places), we look at the third decimal place. Since \(4\lt5\), we keep the second decimal place as it is.

Answer:

\(19.71\)