Question Statement

At a certain time of day, a $20$ meters tall flagpole casts a shadow of $15$ meters. If a nearby building casts a shadow of $50$ meters at the same time, find the height of the building using the inverse tangent function.

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Background and Explanation

This problem uses trigonometry based on the principle that at the same time of day, the sun's rays hit the earth at the same angle ($\theta$). By using the tangent ratio, which relates the height of an object (perpendicular) to its shadow length (base), we can find this angle and use it to determine unknown heights.

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Solution

To solve this problem, we represent the flagpole and the building as two right-angled triangles. Since the measurements are taken at the same time, the angle of elevation of the sun, $\theta$, is identical for both.

1. Identify the given values for the flagpole

Let the flagpole be represented by triangle $ABC$:

• $AB=\text{Height of flagpole}=20\text{ m}$
• $AC=\text{Length of shadow}=15\text{ m}$
• $\angle ACB=\theta$ (Angle of elevation)

2. Calculate the angle of elevation ($\theta$)

Using the tangent ratio for triangle $ABC$: $\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{AB}{AC}$ $\tan \theta =\frac{20}{15}$

To find the angle $\theta$, we use the inverse tangent function: $\theta ={\tan }^{-1}\left(\frac{20}{15}\right)$ $\theta \approx 53.13^{\circ }$

3. Identify the values for the building

Let the building be represented by triangle $PQR$:

• $PQ=\text{Height of building}=y$
• $PR=\text{Length of shadow}=50\text{ m}$
• $\angle PRQ=\theta =53.13^{\circ }$

4. Calculate the height of the building

Using the tangent ratio for triangle $PQR$: $\theta ={\tan }^{-1}\left(\frac{\text{Perp}}{\text{Base}}\right)$ $\theta ={\tan }^{-1}\left(\frac{PQ}{PR}\right)$

Substituting the known values: $\tan 53.13^{\circ }=\frac{PQ}{50}$

Since we know $\tan 53.13^{\circ }=\frac{20}{15}$ from the previous step: $PQ=50\times \frac{20}{15}$ $PQ=50\times \mathrm{1.333...}$ $PQ=66.67\text{ m}$

The height of the building is $66.67$ meters.

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Key Formulas or Methods Used

• Tangent Ratio: $\tan \theta =\frac{\text{Opposite (Perpendicular)}}{\text{Adjacent (Base)}}$
• Inverse Tangent Function: $\theta ={\tan }^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)$
• Similar Triangles: Objects and their shadows form proportional triangles when the light source angle is the same.

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Summary of Steps

1. Use the flagpole's height ($20\text{ m}$) and shadow ($15\text{ m}$) to set up the tangent ratio.
2. Apply the inverse tangent function to find the sun's angle of elevation ($\theta \approx 53.13^{\circ }$).
3. Set up the tangent ratio for the building using the same angle $\theta$ and its shadow length ($50\text{ m}$).
4. Solve for the unknown height ($PQ$) by multiplying the shadow length by the tangent of the angle.