Question

Jessica 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 34 ft from the tree, and Jessica is standing 10.9 ft from the mirror, as shown in the figure. Her eyes are 6 ft above the ground. How tall is the tree? Round your answer to the nearest foot. (The figure is not drawn to scale.)

Answer 1

To find the height of the tree, we can utilize the concept of similar triangles.

1. Define the variables:

   • Let \( h \) be the height of the tree.
   • Jessica's eye level is 6 ft above the ground.
   • The distance from the mirror to the tree is 34 ft.
   • The distance from Jessica to the mirror is 10.9 ft.

2. Set up the triangles: We can create two triangles:

   • Triangle \( A \) comprises Jessica's height (6 ft) and the distance from Jessica to the mirror (10.9 ft).
   • Triangle \( B \) comprises the height of the tree (\( h \)) and the distance from the mirror to the tree (34 ft).

3. Form the ratio of the triangles: The triangles are similar, which means we can write the following proportion: \[ \frac{\text{height of Jessica's eyes}}{\text{distance from Jessica to the mirror}} = \frac{\text{height of the tree}}{\text{distance from the mirror to the tree}} \] Substituting the known values gives: \[ \frac{6}{10.9} = \frac{h}{34} \]

4. Solve for \( h \): Cross-multiplying yields: \[ 6 \cdot 34 = h \cdot 10.9 \] This simplifies to: \[ 204 = 10.9h \] Now, solve for \( h \): \[ h = \frac{204}{10.9} \approx 18.7 \]

5. Round to the nearest foot: Rounding \( 18.7 \) to the nearest foot gives \( 19 \).

Thus, the height of the tree is approximately \( \boxed{19} \) feet.