Question Statement

An architect is designing a roof with a $20^{\circ }$ pitch to allow snow to slide off easily. If the width of the building is $12$ meters, calculate the rise of the roof.

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Background and Explanation

To find the dimensions of a roof, we use right-angled trigonometry. The roof forms a right triangle where:

• The half-width of the building is the adjacent side (base)
• The rise (height) is the opposite side
• The pitch angle is the angle between the slope and the horizontal

Since the roof is symmetric, the half-width $=\frac{12}{2}=6$ m.

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Solution

We identify the following parameters:

• Half-width (adjacent side) $=6$ m
• Rise (opposite side) $=y$
• Pitch angle $\theta =20^{\circ }$

Step 1: Set Up the Trigonometric Ratio

Using the tangent ratio (opposite over adjacent): $\tan 20^{\circ }=\frac{y}{6}$

Step 2: Solve for the Rise

Rearranging: $y=6\tan 20^{\circ }$

Step 3: Calculate

$y=6\times 0.364=2.18\text{ m}$

The rise of the roof is $\approx 2.18$ meters.

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Alternative using inverse trigonometry: If instead we know the slope length $L$ and the rise $y$, we can find the pitch angle: $\theta ={\sin }^{-1}\left(\frac{y}{L}\right)$

For example, if $y=2.18$ m and $L=\frac{6}{\cos 20^{\circ }}\approx 6.39$ m: $\theta ={\sin }^{-1}\left(\frac{2.18}{6.39}\right)={\sin }^{-1}(0.341)\approx 20^{\circ }✓$

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Key Formulas Used

• Tangent Ratio: $\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$
• Inverse Sine: $\theta ={\sin }^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)$

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Summary of Steps

1. Identify the half-width as the adjacent side: $6$ m.
2. Use $\tan 20^{\circ }=\frac{y}{6}$ to find the rise.
3. Calculate $y=6\tan 20^{\circ }\approx 2.18$ m.