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Math  /  Geometry

Question5. A video camera is mounted on top of a 120 m building. When the camera tilts down 36∘36^{\circ} with the horizontal, it views the bottom of another building. If it tilts up 47∘47^{\circ} with the horizontal, it can view the top of the same building. Determine the height of the building viewed by the camera.

Studdy Solution

STEP 1

1. The camera is mounted on top of a 120 m building.
2. The camera tilts down 36∘ 36^\circ to view the bottom of another building.
3. The camera tilts up 47∘ 47^\circ to view the top of the same building.
4. We need to determine the height of the other building.

STEP 2

1. Define the problem using a diagram and identify the right triangles involved.
2. Use trigonometric relationships to find the horizontal distance between the two buildings.
3. Use trigonometric relationships to find the height of the other building.

STEP 3

Draw a diagram with two buildings. The first building is 120 m tall, and the camera is on top of it. The second building is the one we need to find the height of. The camera tilts down 36∘ 36^\circ to view the bottom and tilts up 47∘ 47^\circ to view the top of the other building. Identify the right triangles formed by these angles.

STEP 4

Using the triangle formed by the camera tilting down 36∘ 36^\circ , let d d be the horizontal distance between the two buildings. The tangent of the angle is the opposite side (120 m) over the adjacent side (distance d d ):
tan⁡(36∘)=120d \tan(36^\circ) = \frac{120}{d}
Solve for d d :
d=120tan⁡(36∘) d = \frac{120}{\tan(36^\circ)}

STEP 5

Using the triangle formed by the camera tilting up 47∘ 47^\circ , let h h be the height of the other building. The tangent of the angle is the opposite side (height of the building minus 120 m) over the adjacent side (distance d d ):
tan⁡(47∘)=h−120d \tan(47^\circ) = \frac{h - 120}{d}
Substitute the expression for d d from STEP_2:
tan⁡(47∘)=h−120120tan⁡(36∘) \tan(47^\circ) = \frac{h - 120}{\frac{120}{\tan(36^\circ)}}

STEP 6

Solve for h h :
h−120=tan⁡(47∘)×120tan⁡(36∘) h - 120 = \tan(47^\circ) \times \frac{120}{\tan(36^\circ)}
h=120+tan⁡(47∘)×120tan⁡(36∘) h = 120 + \tan(47^\circ) \times \frac{120}{\tan(36^\circ)}

STEP 7

Calculate the value of h h using a calculator:
h=120+tan⁡(47∘)×120tan⁡(36∘) h = 120 + \tan(47^\circ) \times \frac{120}{\tan(36^\circ)}
h≈120+1.0724×1200.7265 h \approx 120 + 1.0724 \times \frac{120}{0.7265}
h≈120+1.0724×165.2 h \approx 120 + 1.0724 \times 165.2
h≈120+177.2 h \approx 120 + 177.2
h≈297.2 h \approx 297.2
The height of the building viewed by the camera is approximately:
297.2 m \boxed{297.2 \text{ m}}

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