The side of a trinagular roof is to have the same height h and base b^2 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^2h. For a trapezoid, A = b1 + b2/2h.)

b. Compare the ratio from part of question a to what the ratio will be if b1 is doubled for trapezoidal roof and b2 is doubled for both roofs.

(a) want a ratio? Just plug in the numbers for a fraction

(1/2 b^2 h) / ((b1+b2)/2 h)
= b^2/(b1+b2)

For (b), just substitute in new values and see how the ratio changes.

By the way, having b^2 for one figure and b1+b2 for the other plays havoc with the units. Better check your problem again.

a. To find the ratio of the area of the triangular roof to the area of the trapezoidal roof, let's start by calculating the areas of both roofs separately using the given formulas.

For the triangular roof, the area (A) is given by A = 1/2b^2h.

For the trapezoidal roof, the area (A) is given by A = (b1 + b2)/2h.

Let's substitute the values of both roofs using the given information that they have the same height h and base b^2. So, for the trapezoidal roof, we can substitute b1 = b^2 and b2 = b^2.

For the triangular roof, A_triangle = 1/2b^2h.

For the trapezoidal roof, A_trapezoid = (b^2 + b^2)/2h = 2b^2/2h = b^2/h.

Now, we can find the ratio of the areas by dividing the area of the triangular roof by the area of the trapezoidal roof:

Ratio = A_triangle / A_trapezoid = (1/2b^2h) / (b^2/h).

To simplify the ratio, we can cancel out the common factors:

Ratio = (1/2) / 1 = 1/2.

Therefore, the ratio of the area of the triangular roof to the area of the trapezoidal roof is 1/2.

b. Now, let's consider the changes in the ratio if b1 is doubled for the trapezoidal roof and b2 is doubled for both roofs.

If b1 is doubled, the expression becomes A_trapezoid = (2b^2 + b2)/2h.

If b2 is also doubled for both roofs, the expression for the trapezoidal roof becomes A_trapezoid = (2b^2 + 2b^2)/2h = 4b^2/2h = 2b^2/h.

Now, let's recalculate the ratio of the areas:

New Ratio = A_triangle / A_trapezoid = (1/2b^2h) / (2b^2/h) = 1/4.

Therefore, if b1 is doubled for the trapezoidal roof and b2 is doubled for both roofs, the new ratio of the area of the triangular roof to the area of the trapezoidal roof will be 1/4.