Question

heather wants to measure the height of a tree. she sights the top of the tree, using a mirror that is lying flat on the ground. the mirror is 36 ft from the tree, and heather is standing 8.8 ft from the mirror, as shown in the figure. her eyes are 5 ft above the ground. how tall is the tree? round your answer to the nearest foot. (the figure is not drawn to scale.)

Explanation:

Step1: Identify similar triangles

The two triangles (tree - mirror - ground and Heather - mirror - ground) are similar by AA (angle - angle) similarity (right angles and equal angles from reflection).

Step2: Set up proportion

Let \( h \) be the height of the tree. The proportion is \(\frac{h}{36}=\frac{5}{8.8}\) (height over distance from mirror for each triangle).

Step3: Solve for \( h \)

Cross - multiply: \( h=\frac{5\times36}{8.8}=\frac{180}{8.8}\approx20.4545\)

Step4: Round to nearest foot

\( h\approx20 \) (wait, no, wait: \( \frac{5\times36}{8.8}=\frac{180}{8.8}\approx20.45\), but wait, maybe I made a mistake. Wait, 5 is Heather's eye height, 8.8 is her distance from mirror, 36 is tree's distance from mirror. So the correct proportion is \(\frac{h}{36}=\frac{5}{8.8}\), so \( h = \frac{5\times36}{8.8}=\frac{180}{8.8}\approx20.45\), but when we calculate \( 5\times36 = 180\), \( 180\div8.8\approx20.45\), which rounds to 20? Wait, no, 8.8 times 20 is 176, 8.8 times 21 is 184.8. Wait, 180 is between 176 and 184.8. Wait, maybe I messed up the proportion. Wait, the two triangles: one has height 5 and base 8.8, the other has height \( h \) and base 36. So similarity ratio: \(\frac{h}{5}=\frac{36}{8.8}\). Oh! I had the proportion reversed. That's the mistake. So correct proportion: \(\frac{h}{5}=\frac{36}{8.8}\). Then \( h=\frac{5\times36}{8.8}=\frac{180}{8.8}\approx20.45\)? Wait, no, \(\frac{h}{36}=\frac{5}{8.8}\) is the same as \(\frac{h}{5}=\frac{36}{8.8}\) because cross - multiplying gives \( 8.8h = 5\times36\) in both cases. Wait, \( 5\times36 = 180\), \( 180\div8.8\approx20.45\), which is approximately 20 when rounded to the nearest foot? Wait, no, 20.45 is closer to 20? Wait, 0.45 is less than 0.5, so yes, 20. But wait, let's recalculate: \( 8.8\times20 = 176\), \( 5\times36 = 180\), \( 180 - 176 = 4\), \( 8.8 - 4 = 4.8\)? No, wait, the formula is correct. So the height of the tree is approximately 20 feet? Wait, no, maybe I made a mistake in the proportion. Let's think again. The two right triangles: one with vertical side 5 (Heather's height) and horizontal side 8.8 (her distance from mirror), the other with vertical side \( h \) (tree's height) and horizontal side 36 (tree's distance from mirror). Since they are similar, the ratios of corresponding sides are equal. So \(\frac{h}{36}=\frac{5}{8.8}\) (vertical over horizontal for each triangle). So \( h=\frac{5\times36}{8.8}=\frac{180}{8.8}\approx20.45\), which rounds to 20. But let's check with the other ratio: \(\frac{h}{5}=\frac{36}{8.8}\), \( h=\frac{5\times36}{8.8}\approx20.45\). So the correct height is approximately 20 feet when rounded to the nearest foot. Wait, but 20.45 is closer to 20 than 21? Wait, 0.45 is less than 0.5, so yes, rounds to 20.

Answer:

\( \boxed{20} \) (Wait, but when I recalculate \( 180\div8.8 \): 8.8 × 20 = 176, 180 - 176 = 4, 4÷8.8≈0.45, so total is 20.45, which is 20 when rounded to the nearest foot. But maybe the problem expects a different approach? Wait, no, the key is the similar triangles. So the correct answer after correcting the proportion (initially reversed, but then realized that both ways give the same equation) is approximately 20.