Question

1. a studio is 73m to the left of a school. the angle of elevation from the base of the studio to the roof of the school is 44°. the angle of depression from the roof of the studio to the roof of the school is 79°.

a find the height of the school to 3 decimal places:
b how much higher is the studio than the school to 3 decimal places?
c what is the total height of the studio to 1 decimal place?

Explanation:

Part (a)

Step1: Identify the right triangle for the school's height

We have a right triangle with the adjacent side (distance between studio and school) as \( 73 \, \text{m} \) and the angle of elevation \( 44^\circ \). The height of the school \( s \) can be found using the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). So, \( \tan(44^\circ)=\frac{s}{73} \).

Step2: Solve for \( s \)

Multiply both sides by \( 73 \): \( s = 73\times\tan(44^\circ) \). Calculate \( \tan(44^\circ)\approx0.9656887748 \), then \( s = 73\times0.9656887748\approx70.495 \).

Step1: Identify the right triangle for the height difference \( h \)

For the height difference \( h \) between the studio and the school, we use the angle of depression (which is equal to the angle of elevation) \( 79^\circ \). The adjacent side is still \( 73 \, \text{m} \), so \( \tan(79^\circ)=\frac{h}{73} \).

Step2: Solve for \( h \)

Multiply both sides by \( 73 \): \( h = 73\times\tan(79^\circ) \). Calculate \( \tan(79^\circ)\approx5.144554224 \), then \( h = 73\times5.144554224\approx375.552 \)? Wait, no, wait. Wait, the angle of depression from the studio's roof to the school's roof is \( 79^\circ \), so the height difference \( h \) is \( 73\times\tan(79^\circ) \)? Wait, no, let's re - check. Wait, the angle of depression is from the studio's roof to the school's roof. So the horizontal distance is \( 73 \, \text{m} \), and the height difference \( h \) is given by \( \tan(79^\circ)=\frac{h}{73} \), so \( h = 73\times\tan(79^\circ) \). Wait, but let's calculate \( \tan(79^\circ) \): \( \tan(79^\circ)\approx5.14455 \), so \( 73\times5.14455 = 73\times5+73\times0.14455=365 + 10.55215 = 375.55215 \)? Wait, that can't be right. Wait, no, maybe I mixed up the angle. Wait, the angle of elevation from the school's roof to the studio's roof is \( 79^\circ \), so the height difference \( h \) is \( 73\times\tan(79^\circ) \). Wait, but let's compute it correctly. \( \tan(79^\circ)\approx5.144554224 \), so \( 73\times5.144554224 = 73\times5.144554224 \). Let's calculate: \( 5\times73 = 365 \), \( 0.144554224\times73=10.55245835 \), so total is \( 365 + 10.55245835 = 375.55245835 \). Wait, but that seems too big. Wait, maybe the angle is \( 79^\circ \), but let's check the diagram again. Wait, the angle of depression from the studio's roof to the school's roof is \( 79^\circ \), so the height difference \( h \) is the opposite side, adjacent is \( 73 \, \text{m} \). So \( \tan(79^\circ)=\frac{h}{73} \), so \( h = 73\tan(79^\circ) \approx 73\times5.14455\approx375.552 \). Then, the height of the studio is \( s + h \)? No, wait, no. Wait, the school's height is \( s \), and the studio is taller by \( h \). Wait, no, the angle of elevation from the base of the studio to the roof of the school is \( 44^\circ \) (which gives the school's height), and the angle of depression from the roof of the studio to the roof of the school is \( 79^\circ \), which gives the height difference between the studio and the school. So the height difference \( h \) is \( 73\tan(79^\circ) \approx 73\times5.14455\approx375.552 \)? That seems very large. Wait, maybe I made a mistake in the angle. Wait, maybe the angle is \( 79^\circ \), but let's check the tangent of \( 79^\circ \). Yes, \( \tan(79^\circ)\approx5.14455 \). So \( 73\times5.14455 = 375.552 \). Then, the height of the studio is \( s + h \), but the question is "how much higher is the studio than the school", which is \( h \). Wait, but that seems too big. Wait, maybe the distance is 7.3m? No, the problem says 73m. Wait, maybe the angle is \( 7.9^\circ \)? No, the diagram shows \( 79^\circ \). Wait, maybe I misread the angle. Wait, the problem says "the angle of depression from the roof of the studio to the roof of the school is \( 79^\circ \)". So the horizontal distance is 73m, so the height difference \( h \) is \( 73\tan(79^\circ) \approx 375.552 \). Then, the height of the studio is \( s+h = 70.495+375.552 = 446.047 \), but that seems very tall. Wait, maybe the angle is \( 19^\circ \) instead of \( 79^\circ \)? No, the diagram shows \( 79^\circ \). Wait, maybe I made a mistake in the tangent…

(corrected approach):
Wait, no! Wait a second. The angle of depression from the studio's roof to the school's roof is \( 79^\circ \), which means that the angle between the horizontal line from the studio's roof and the line of sight to the school's roof is \( 79^\circ \). So the height difference \( h \) is the opposite side, and the adjacent side is 73m. So \( \tan(79^\circ)=\frac{h}{73} \), so \( h = 73\tan(79^\circ) \approx 73\times5.14455\approx375.552 \). But this seems very large. Alternatively, maybe the angle is \( 7.9^\circ \), but the problem says \( 79^\circ \). Maybe the distance is 7.3m? If we take 7.3m, then \( h = 7.3\tan(79^\circ)\approx7.3\times5.14455\approx37.555 \, \text{m} \), which is more reasonable. But the problem says 73m. So perhaps there is a mistake in the problem, but according to the given data, we proceed. So the height difference is \( 73\tan(79^\circ)\approx375.552 \, \text{m} \).

Answer:

The height of the school is \( \boldsymbol{70.495} \, \text{m} \) (to 3 decimal places).

Part (b)