Ella is building a house in Montreal. The roof of the house must have a minimum slope (pitch). This is to allow rain and snow to slide off the roof more easily. The pitch of Ella's roof is "6 in 12". She draws the diagram below to show her house, and marks the angle $\beta$. (a) (i) Using the right-angled triangle $ABC$, write down the value of $\tan \beta$ as a fraction. (ii) Calculate the measure of the angle $\beta$. Give your answer correct to the nearest degree. (b) Find the length of the roof from A to B. Give your answer correct to two decimal places.

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Ella is building a house in Montreal - Junior Cycle Mathematics - Question 13 - 2014

Question 13

Ella is building a house in Montreal. The roof of the house must have a minimum slope (pitch). This is to allow rain and snow to slide off the roof more easily. The ... show full transcript

Worked Solution & Example Answer:Ella is building a house in Montreal - Junior Cycle Mathematics - Question 13 - 2014

Step 1

Using the right-angled triangle ABC, write down the value of tan β as a fraction

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Answer

In triangle $ABC$ , the tangent of angle $\beta$ is defined as the ratio of the opposite side to the adjacent side. Therefore, we have:

$\tan \beta = \frac{BC}{AC} = \frac{6}{12} = \frac{1}{2}$

Step 2

Calculate the measure of the angle β. Give your answer correct to the nearest degree.

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Answer

To find the angle $\beta$ , we use the inverse tangent function:

$\beta = \tan^{-1}(0.5) \approx 26.57^{\circ}$

Rounding to the nearest degree gives:

$\beta \approx 27^{\circ}$

Step 3

Find the length of the roof from A to B. Give your answer correct to two decimal places.

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Answer

Using the Pythagorean theorem on triangle $ABC$ , we can calculate the length of $|AB|$ as follows:

$|AB|^2 = |AC|^2 + |BC|^2$ $|AB|^2 = 6^2 + 12^2$ $|AB|^2 = 36 + 144 = 180$

Taking the square root gives:

$|AB| = \sqrt{180} \approx 13.42$

Thus, the length of the roof from A to B is approximately 13.42 m.

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