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.
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.
To solve part a of the question, we need to find the ratio of the area of the triangular roof to the area of the trapezoidal roof. This ratio can be calculated by dividing the area of the triangular roof by the area of the trapezoidal roof.
The area of a triangle is given by the formula A = (1/2) * b * h, where b is the base length and h is the height. In this case, both the base length and height of the triangular roof are denoted as b₂ and h.
The area of a trapezoid is given by the formula A = (b₁ + b₂) / 2 * h, where b₁ and b₂ are the parallel base lengths and h is the height. In this case, the base length of the trapezoidal roof is denoted as b₁ and the other side matches the base length and height of the triangular roof (b₂ and h).
Let's substitute the given values into the formulas and calculate the ratio of the two areas.
For the triangular roof:
A_triangle = (1/2) * b₂ * h
For the trapezoidal roof:
A_trapezoidal = (b₁ + b₂) / 2 * h
Now, we can find the ratio A_triangle / A_trapezoidal:
Ratio = (A_triangle) / (A_trapezoidal)
= (1/2 * b₂ * h) / ((b₁ + b₂) / 2 * h)
= (1/2 * b₂ * h) * (2 * h / (b₁ + b₂))
= (b₂ * h^2) / (b₁ + b₂)
Therefore, the ratio of the area of the triangular roof to the area of the trapezoidal roof is (b₂ * h^2) / (b₁ + b₂).
To solve part b of the question, we need to compare the ratio from part a with the new ratio when b₁ is doubled for the trapezoidal roof and b₂ is doubled for both roofs.
Let's denote the new values as b₁' = 2b₁ and b₂' = 2b₂.
The new ratio, let's call it Ratio', is given by:
Ratio' = (b₂' * h^2) / (b₁' + b₂')
= (2b₂ * h^2) / (2b₁ + 2b₂)
= (b₂ * h^2) / (b₁ + b₂)
= Ratio
Therefore, if both b₁ and b₂ are doubled for the trapezoidal and triangular roofs, the ratio of their areas remains the same.
I hope this explanation helps you understand how to solve both parts of the question!