Question Statement

The roof of a warehouse needs to be designed with a $30^{\circ }$ incline to ensure water runoff. If the building is 20 meters wide, determine the rise of the roof using the inverse sine function.

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Background and Explanation

To solve this problem, we use right-angle trigonometry. By splitting the symmetrical roof into two right-angled triangles, we can use the sine and cosine ratios to determine the unknown height (rise) based on the given width and angle of inclination.

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Solution

Based on the provided diagram:

1. Identify the known values

The total width of the building is given as $CD=20\text{ m}$. The roof forms a triangle where the base of the right-angled portion ($AC$) is half of the total width: $AC=\text{Base of building}=\frac{CD}{2}=\frac{20}{2}=10\text{ m}$

The angle of inclination is: $\angle ACB=30^{\circ }=\theta$

The height (rise) we need to find is: $AB=\text{Height of building}=y$

2. Calculate the Hypotenuse ($BC$)

We use the cosine ratio because we know the adjacent side (base) and the angle: $\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}=\frac{10}{BC}$ $\cos 30^{\circ }=\frac{10}{BC}$

Since $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$: $\frac{\sqrt{3}}{2}=\frac{10}{BC}$ $BC=\frac{10\times 2}{\sqrt{3}}=\frac{20}{\sqrt{3}}\text{ m}$

3. Calculate the Rise ($y$)

Now, we use the sine ratio to find the perpendicular height ($y$): $\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{y}{BC}$ $\sin 30^{\circ }=\frac{y}{\frac{20}{\sqrt{3}}}$

Rearranging the equation to solve for $y$: $\frac{20}{\sqrt{3}}\sin 30^{\circ }=y$ $y=\frac{20}{\sqrt{3}}\times \frac{1}{2}$ $y=\frac{10}{\sqrt{3}}$

Converting to a decimal value: $y\approx 5.77\text{ m}$

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Key Formulas or Methods Used

• Base Calculation: $\text{Base}=\frac{\text{Total Width}}{2}$
• Cosine Ratio: $\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Sine Ratio: $\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$
• Trigonometric Values: $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$ and $\sin 30^{\circ }=\frac{1}{2}$

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Summary of Steps

1. Divide the total building width (20m) by 2 to find the base of the right triangle (10m).
2. Use the cosine of $30^{\circ }$ to calculate the length of the roof slant (hypotenuse $BC$).
3. Apply the sine ratio using the hypotenuse and the $30^{\circ }$ angle to find the vertical rise $y$.
4. Simplify the fraction $\frac{10}{\sqrt{3}}$ to get the final height of approximately $5.77\text{ m}$.