Question

From the stage of a theater, the angle of elevation to the first balcony is 19 degrees. The angle of elevation to the second balcony, 6.3 meters directly above the first, is 29 degrees. How high above stage level is the first balcony, to the nearest tenth of a meter

Answer 1

The first balcony is about 10.3 meters high above stage level.

Step-by-step explanation:

To solve this problem, we will use trigonometric principles, particularly the tangent function, since we are dealing with right triangles formed by the line of sight to the balconies and the heights and distances we want to find.

Let's label the height of the first balcony as 'h' meters. The distance from the point directly below the first balcony on the stage to the observer is unknown; let's call this 'd' meters.

For the first balcony:
We are given the angle of elevation θ₁ to the first balcony is 19 degrees. Assuming the theater stage is level, we have a right triangle where 'h' is opposite to the angle, and 'd' is adjacent to the angle. We can write the tangent function as follows:

tan(θ₁) = opposite/adjacent
tan(19°) = h/d
h = d * tan(19°) …(1)

For the second balcony:
The angle of elevation θ₂ to the second balcony is 29 degrees, and it's 6.3 meters above the first balcony, which means the total height of the second balcony from the stage is 'h + 6.3 meters'.

Using the same point below the first balcony as our adjacent side (distance 'd' meters), and 'h + 6.3 meters' as our opposite side, we write another equation using the tangent function for the second balcony:

tan(θ₂) = opposite/adjacent
tan(29°) = (h + 6.3)/d
(h + 6.3) = d * tan(29°) …(2)

Now we have two equations with two unknowns (h and d). We can solve these equations by expressing 'd' from equation (1) and substituting into equation (2).

From equation (1):
d = h / tan(19°)

Substitute 'd' from equation (1) into equation (2):
h + 6.3 = (h / tan(19°)) * tan(29°)

Now we solve for 'h':
h + 6.3 = h * (tan(29°) / tan(19°))
6.3 = h * (tan(29°) / tan(19°)) - h
6.3 = h * ((tan(29°) / tan(19°)) - 1)

Isolating 'h':
h = 6.3 / ((tan(29°) / tan(19°)) - 1)

Now plug in the values of the tangent functions using a calculator:
h = 6.3 / ((tan(29) / tan(19)) - 1)

Carrying out the calculation:
h ≈ 6.3 / ((0.554309051 / 0.344327613) - 1)
h ≈ 6.3 / (1.6098472 - 1)
h ≈ 6.3 / 0.6098472
h ≈ 10.33 meters

Therefore, the height of the first balcony above the stage level is approximately 10.3 meters to the nearest tenth of a meter.

Answer 2

10.3 meters.

Step-by-step explanation:

From Triangle ABC

`\tan 29^\circ =(6.3+x)/(h) \\h \tan 29^\circ=6.3+x\\h=(6.3+x)/(\tan 29^\circ)`

From Triangle ADC

`\tan 19^\circ =(x)/(h) \\h \tan 19^\circ=x\\h=(x)/(\tan 19^\circ)`

Since the values of h are the same

`(x)/(\tan 19^\circ)=(6.3+x)/(\tan 29^\circ)\\\\x\tan 29^\circ=\tan 19^\circ(6.3+x)\\x\tan 29^\circ=6.3\tan 19^\circ+x\tan 19^\circ\\x\tan 29^\circ-x\tan 19^\circ=6.3\tan 19^\circ\\x(\tan 29^\circ-\tan 19^\circ)=6.3\tan 19^\circ\\x=(6.3\tan 19^\circ)/(\tan 29^\circ-\tan 19^\circ) \\x=10.3$ meters (to the nearest tenth of a meter)`

The height of the first balcony above stage level is 10.3 meters.