Question

If the height of a gable end of a roof is 7.5 m and the rafters are 10.2 long, at what angle do the rafters slope, and how wide is the gable end at the base?

Answer 1

Solution:

Consider the following diagram of the problem situation:

We need to find the angle alpha and the base b. To find the angle alpha, we can apply the following trigonometric identity:

$\sin (\alpha)=\frac{h}{l}$

Replacing the data of the problem in the above equation, we get:

$\sin (\alpha)=\frac{7.5}{10.2}$

now, applying the inverse function of the sine function, we get:

$\alpha=\sin ^{-1}(\frac{7.5}{10.2})=\text{ 47.33}$

then, the angle would be:

$\alpha=47.33^{\circ}$

On the other hand, to find the lenght of the base, we can apply the Pythagorean Theorem:

$b\text{ = }\sqrt[]{l^2-h^2}\text{ = }\sqrt[]{(10.2)^2-(7.5)^2}\text{ = 6.91}$

so that, we can conclude that the base measures:

$b\text{ = 6.91}$