Explanation:
Let's call the shorter base of the trapezoid "b1", the longer base "b2", and the height "h". We can use the formula for the surface area of a right prism to set up an equation:
Surface area of prism = 2(base area) + (lateral area) = 1280
The base area is the area of the trapezoid, which is given by:
(base area) = (1/2)(b1 + b2)h
The lateral area is the area of the four rectangular faces of the prism, which are all congruent. Each face has an area equal to the product of the height and the length of one of the legs of the trapezoid, so the lateral area is:
(lateral area) = 4hl
where l is the length of one of the legs of the trapezoid.
Substituting these expressions into the formula for the surface area of the prism, we get:
2[(1/2)(b1 + b2)h] + 4hl = 1280
Simplifying and rearranging, we get:
h(b1 + b2) + 2hl = 1280
We also know that the sum of the lengths of the legs of the trapezoid is 52 feet, which means:
l1 + l2 = 52
But we can express l1 and l2 in terms of b1 and b2 using the formula for the area of a trapezoid:
(base area) = (1/2)(b1 + b2)h = (1/2)(l1 + l2)h
Simplifying, we get:
b1 + b2 = (l1 + l2)h
Substituting this into the previous equation, we get:
h[(l1 + l2)h] + 2hl = 1280
Simplifying, we get:
h^2(l1 + l2) + 2hl = 1280
Substituting l1 + l2 = 52, we get:
h^2(52) + 2hl = 1280
This is a quadratic equation in h. We can solve it using the quadratic formula:
h = [-2l ± sqrt(4l^2 + 4h^2(52)(1280 - 2hl))] / 2(52)
Simplifying and factoring out a 2, we get:
h = [-l ± sqrt(l^2 + h^2(1280 - 2hl))] / 52
We have two possible solutions for h, but one of them is negative, which doesn't make sense in the context of the problem. So we can discard the negative solution and focus on the positive one:
h = [-l + sqrt(l^2 + h^2(1280 - 2hl))] / 52
We don't know the exact value of h yet, but we can use this equation to set up a system of equations that we can solve for h. Specifically, we can use the fact that the legs of the trapezoid add up to 52 feet to solve for l in terms of b1 and b2:
l = 52 - (b1 + b2)
Substituting this into the equation for h, we get:
h = [-l + sqrt(l^2 + h^2(1280 - 2hl))] / 52
h = [-52 + (b1 + b2) + sqrt((52 - (b1 + b2))^2 + h^2(1280 - 2h(b1 + b