Question Statement

A surveyor is standing 50 meters away from a building and measures the angle of elevation to the top of the building as $35^{\circ }$. What is the height of the building? Use the inverse tangent function to find the height.

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Background and Explanation

This problem utilizes right-angle trigonometry to find an unknown height. By representing the scenario as a right triangle, we can use the tangent ratio, which relates the angle of elevation to the opposite side (height) and the adjacent side (distance).

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Solution

To solve for the height of the building, we first identify the components of the right triangle formed by the surveyor and the building:

• $AB$: Height of the building (represented as $y$)
• $AC$: Distance of the surveyor from the building ($50\text{ m}$)
• $\theta$: Angle of elevation ($35^{\circ }$)

The relationship between the angle and the sides is defined by the inverse tangent function:

$\theta ={\tan }^{-1}\left(\frac{AB}{AC}\right)$

By taking the tangent of both sides, we get the standard tangent ratio:

$\tan \theta =\frac{AB}{AC}$

Now, we substitute the known values into the equation:

$\tan 35^{\circ }=\frac{AB}{50}$

To isolate $AB$ (the height), we multiply both sides by 50:

$AB=50\times \tan 35^{\circ }$

Using a calculator to find $\tan 35^{\circ }\approx 0.7002$:

$AB\approx 35.01\approx 35\text{ m}$

The height of the building is approximately $35\text{ meters}$.

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Key Formulas or Methods Used

• Tangent Ratio: $\tan (\theta )=\frac{\text{Opposite}}{\text{Adjacent}}$
• Inverse Tangent: $\theta ={\tan }^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$
• Algebraic Rearrangement: Multiplying by the denominator to solve for the unknown side.

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Summary of Steps

1. Identify the height as the opposite side ($AB$) and the distance as the adjacent side ($AC$).
2. Set up the trigonometric relationship using $\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$.
3. Substitute the given angle ($35^{\circ }$) and the distance ($50\text{ m}$) into the formula.
4. Solve for $AB$ by multiplying the distance by the tangent of the angle.
5. Calculate the final result and round to the nearest whole number.