Question

The side of a trinagular roof is to have the same height h and base b (with a 2 below) as the side of a trapezodial roof.

a. What is the ratio of the area of the triangular roof to the area of the trapezoidal roof?

(For a triangle, A = 1/2b(with a 2 below)h. For a trapezoid, A = b (with a 1 below) + b (with a 2 below)/2h.)

b. Compare the ratio from part of question a to what the ratio will be if b (with a 1 below) is doubled for trapezoidal roof and b (with a 2 below) is doubled for both roofs.

Answer 1

The question still is poorly worded. What is the "side" of a roof? I expect we are looking at the end of the building, at the cross-section area.

The triangle has area a1=b2*h/2
The trapezoid has area a2=(b1+b2)*h/2

The ratio of areas is thus
a1/a2 = (b2*h/2) / (b1+b2)*h/2 = b2/(b1+b2)

If both b's are doubled, then we have

(2*b2)/(2*b1+2*b2) = b2/(b1+b2)
The ratio is unchanged.