A metal feeding trough is in the shape of a triangular prism with one of the faces missing. The base of the trough is an equilateral triangle with sides of length 3.5 feet and a height of 4 feet. If the trough is 15 feet long, how much metal was used to construct the trough?
A. 66.5 ft.^2
B. 119 ft.^2
C. 171 ft.^2
D. 194 ft.^2
First, we need to find the area of the equilateral triangle base of the trough. We can do this by using the formula for the area of a triangle: $A = \frac{1}{2}bh$.
We already know the height of the triangle (4 feet) and the base (3.5 feet), so we can plug these values into the formula:
$A = \frac{1}{2}(3.5)(4)$
$A = 7$ square feet
Now we need to find the surface area of the trough. Since one face is missing, there are only four faces we need to find the area of: three triangular sides and the bottom triangle.
The surface area of the trough is the sum of the areas of the four faces:
$SA = A_{base} + 3A_{triangles}$
We already found $A_{base}$ (7 square feet). Now we need to find the area of one of the triangular sides of the trough.
The triangular side has a base of 3.5 feet (the same as the base of the triangular prism) and a height of 15 feet (the length of the trough).
Using the triangle area formula again:
$A_{triangle} = \frac{1}{2}(3.5)(15)$
$A_{triangle} = 26.25$ square feet
Now plug this value back into the surface area formula:
$SA = 7 + 3(26.25)$
$SA = 7 + 78.75$
$SA = 85.75$ square feet
So the surface area of the trough, which represents the amount of metal used to construct it, is 85.75 square feet.
However, this answer is not in the choices. Checking the answer choices, one could guess that the problem was intended to have a trough with a trapezoidal base (typical in feeding troughs) instead of an equilateral triangle. Since it is the only possible way to make sense of the given answer choices, we'll try to solve for this (since it is the only possible way to make sense out of the given answer choices):
If the base of the trough is a trapezoid, we need to find the area of the trapezoid base. The area of a trapezoid is given by the formula:
$A_{trapezoid} = \frac{1}{2}(b_1 + b_2)(h)$
In this case, the two bases of the trapezoid are 3.5 feet (the length of the equilateral triangle sides) and the height is 4 feet.
$A_{trapezoid} = \frac{1}{2}(3.5 + 3.5)(4)$
$A_{trapezoid} = 14$ square feet
Now we need to find the area of one of the triangular sides of the trough. These triangles have a base of 4 feet (the height of the trapezoid) and a height of 15 feet (the length of the trough).
$A_{triangle} = \frac{1}{2}(4)(15)$
$A_{triangle} = 30$ square feet
Now, we need to find the surface area of the trough, consisting of two triangular sides, one trapezoidal base, and one rectangular side (the top of the trough):
$SA = A_{trapezoid} + 2A_{triangles} + A_{rectangle}$
We need to find the area of the rectangle, which has a length of 15 feet (the length of the trough) and a width equal to the difference in the lengths of the top and bottom sides of the base trapezoid, which is $3.5-1.5=2$ feet.
$A_{rectangle} = lw = (2)(15) = 30$ square feet
Plug these values back into the surface area formula:
$SA = 14 + 2(30) + 30$
$SA = 14 + 60 + 30$
$SA = 104$ square feet
So if the trough had a trapezoidal base and a rectangular top, the amount of metal used to construct it would be 104 square feet, which is slightly closer to answer choice D, but since there seemed to be confusion in the original problem, you could explain the situation to your teacher and ask for clarification on the intended problem.