Raising The Roof
How far should the roof overhang to shade windows from the mid-day sun?
Age
14 to 16
Challenge level
Problem
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The diagram shows a cross-sectional view of a house.
The roof space is 12 metres across and 1.5 metres high at the
highest point in the centre.
Find the angle of slope of the roof.
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Also find the sloping distance $b$ from the top of the roof to
the top of the walls.
The hottest part of the day is when the sun is directly
overhead. The roof overhangs by a distance $a$ and it shades all
the windows from the sun while the sun's rays are inclined at 8
degrees or less to the vertical. The lowest part of the windows is
one metre above the ground and the angle $\phi$ in the diagram is 8
degrees. Find the length of the overhang, $a$.
Getting Started
Just simple trigonometry.
Student Solutions
We have had a number of excellent solutions to this problem from Year 11 pupils in Sharnbrook School in Bedforshire. And they didn't all solve the problem in the same way...
Jennie, Emily, Peter and Hannah sent in this solution, and used Pythagoras' Theorem to find one of their answers.
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Megan, Shivani, Will, Austen and Michael managed to find b in a different way, using what they knew about the angles in the roof.
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Mollie, Meg and Jack also used Pythagoras, and relabeled the angles and sides to make it easier for them to answer the questions. Perhaps they should have returned to the original labels at the end of their solution...
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Olivia, Alex and Toby sent in very clear solutions to the first and third parts of the problem. We'll assume that they could have also done the second part!
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Teachers' Resources
Just simple trigonometry.