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Raising the Roof

Age 14 to 16 Challenge Level:

house
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.

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, $a$.