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Chapter 10: Problem 40

From a point on level ground 30 yards from the base of a building, the angle of elevation to the top of the building is \(38.7^{\circ}\). Approximate the height of the building to the nearest foot.

Short Answer

Expert verified
The height of the building, to the nearest foot, is calculated to be about 37 feet.

Step by step solution

01

Identify the given quantities

The problem states that the distance from the base of the building (the adjacent side in our right triangle) is 30 yards, and the angle of elevation is \(38.7^{\circ}\). The task is to find the height of the building (the opposite side of the right triangle).
02

Apply the tangent function

Since tangent of an angle in a right triangle is defined as the ratio of the opposite side (height of building) to the adjacent side (distance from building), we can formulate the problem as: \(\tan(38.7^{\circ}) = \frac{Opposite Side}{Adjacent Side}\). Substituting the given values, we have: \(\tan(38.7^{\circ}) = \frac{height of the building}{30}\).
03

Solve for the height of the building

To solve for the height of the building, we can rearrange our equation and multiply both sides by 30. This gives us: \(Height of the building = 30 \cdot \tan(38.7^{\circ})\)
04

Compute the height

Now, calculate the value using the tangent of 38.7 degrees. Ensure that the calculator is set to degrees.
05

Round to the nearest foot

Once the height is calculated, round it to the nearest foot as the problem indicates. Since 1 yard equals 3 feet, the value should be converted to feet by multiplying by 3 before rounding.

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