Question

in the diagram below, a car at point a is driving toward a building. the vertical height of the building is 84 feet. the angle of elevation from point a to the top of the building is 22°. after moving to point b, the angle of elevation becomes 31°. find, to the nearest foot, the distance traveled from point a to point b. show your work here.

Explanation:

Step1: Find distance from A to building base

Let \( d_A \) be distance from A to building base. Using \( \tan(22^\circ)=\frac{84}{d_A} \), so \( d_A = \frac{84}{\tan(22^\circ)} \). Calculate \( \tan(22^\circ)\approx0.4040 \), then \( d_A\approx\frac{84}{0.4040}\approx207.92 \) feet.

Step2: Find distance from B to building base

Let \( d_B \) be distance from B to building base. Using \( \tan(31^\circ)=\frac{84}{d_B} \), so \( d_B = \frac{84}{\tan(31^\circ)} \). Calculate \( \tan(31^\circ)\approx0.6009 \), then \( d_B\approx\frac{84}{0.6009}\approx139.79 \) feet.

Step3: Find distance from A to B

Distance \( AB = d_A - d_B \approx207.92 - 139.79 = 68.13 \approx 68 \) feet.

Answer:

The distance traveled from point A to point B is approximately \(\boxed{68}\) feet.