Question Statement

In constructing a roof, the horizontal span is 10 meters, and the desired height of the roof's peak is 3 meters. Determine the angle at which the roof trusses should be installed using an inverse cosine function.

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Background and Explanation

To solve this problem, we treat the roof structure as a right-angled triangle. We use the Pythagorean theorem to find the length of the unknown side and then apply the inverse cosine function (${\cos }^{-1}$), which relates the adjacent side (base) and the hypotenuse to find the angle of inclination.

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Solution

Based on the provided diagram and data:

First, we define the dimensions of the triangle formed by the roof:

• Let $AB$ be the peak (height) of the roof: $AB=3\text{ m}$
• Let $BC$ be the span (hypotenuse) of the roof: $BC=10\text{ m}$
• Let $AC$ be the horizontal length (base): $AC=x$
• Let the angle of installation be: $\angle ACB=\theta$

Step 1: Find the horizontal length ($x$)

Using the Pythagoras theorem ($a^2+b^2=c^2$), where $BC$ is the hypotenuse:

$\begin{array}{rl}(BC{)}^2& =(AC{)}^2+(AB{)}^2\\ (10{)}^2& =x^2+3^2\\ 100& =x^2+9\\ x^2& =100-9\\ x^2& =91\\ x& =\sqrt{91}\end{array}$

Step 2: Calculate the angle ($\theta$)

We use the cosine ratio, which is defined as the ratio of the adjacent side (Base) to the hypotenuse (Hyp). To find the angle, we use the inverse cosine function:

$\begin{array}{rl}\theta & ={\cos }^{-1}\left(\frac{\text{Base}}{\text{Hyp}}\right)\\ \theta & ={\cos }^{-1}\left(\frac{\sqrt{91}}{10}\right)\end{array}$

By calculating the value: $\theta \approx 17^{\circ }27^{\prime }$

The angle at which the roof trusses should be installed is approximately $17^{\circ }27^{\prime }$.

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Key Formulas or Methods Used

• Pythagorean Theorem: $c^2=a^2+b^2$ (where $c$ is the hypotenuse)
• Cosine Ratio: $\cos (\theta )=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Inverse Cosine Function: $\theta ={\cos }^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)$

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Summary of Steps

1. Identify the given sides: Height ($3\text{ m}$) and Hypotenuse ($10\text{ m}$).
2. Apply the Pythagorean theorem to solve for the missing horizontal base ($x=\sqrt{91}$).
3. Set up the cosine ratio using the base and the hypotenuse.
4. Use the inverse cosine function to solve for the angle $\theta$.
5. Convert the decimal result into degrees and minutes.