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?
66.5 ft
119 ft
171.5 ft
194 ft
It sounds like the end sections are a triangle mounted by a rectangle. If so, then the ends each have area
4 * 3.5 + 3.5^2/4 √3
the sides are
two rectangles each 3.5*15
two rectangles each 4*15
so add up all the areas
If I got the shape wrong, it's because an equilateral triangle of side 3.5 cannot have a height of 4.
To calculate the amount of metal used to construct the trough, we need to find the surface area of the three sides and the base.
The missing face is a rectangle with dimensions 15 feet (length) and 4 feet (height). The area of this face is 15 feet * 4 feet = 60 square feet.
Each of the two triangular faces is an equilateral triangle with sides of length 3.5 feet. The formula to calculate the area of an equilateral triangle is A = (√3/4) * s^2, where s is the length of the side.
So, the area of each triangular face is (sqrt(3)/4) * (3.5 feet)^2 = (sqrt(3)/4) * 12.25 square feet = 1.3229 square feet.
To find the surface area of the two triangular faces, we multiply the area of one face by 2: 1.3229 square feet * 2 = 2.6458 square feet.
Finally, to find the total surface area of the trough, we add the area of the missing face and the area of the two triangular faces: 60 square feet + 2.6458 square feet = 62.6458 square feet.
Therefore, the amount of metal used to construct the trough is approximately 62.6458 square feet, rounded to the nearest whole number, which is 63 feet.
None of the provided answer choices match the calculated value.
To find the amount of metal used to construct the trough, we need to calculate the surface area of the trough. The surface area consists of the lateral faces (sides) and the base.
First, let's calculate the area of the base of the trough, which is an equilateral triangle. The formula to find the area of an equilateral triangle is:
Area = (sqrt(3) / 4) * side^2
Given that the side length of the equilateral triangle base is 3.5 feet, we can substitute this value into the formula:
Area = (sqrt(3) / 4) * 3.5^2
Area ≈ 5.301 sq ft
Next, let's calculate the area of the lateral faces (sides) of the prism. The lateral faces are three rectangles, each with a length of 15 feet (same as the length of the trough) and a width of 4 feet (same as the height of the trough).
Area of one lateral face = length * width
= 15 ft * 4 ft
= 60 sq ft
Since there are three lateral faces, the total area of all three faces is:
Total lateral area = 3 * 60 sq ft
= 180 sq ft
Finally, to find the total surface area, we need to add the area of the base and the total lateral area:
Total surface area = base area + total lateral area
= 5.301 sq ft + 180 sq ft
≈ 185.301 sq ft
Therefore, the amount of metal used to construct the trough is approximately 185.301 square feet. However, the answer choices provided are in feet, not square feet.
Out of the given answer choices, the closest option to 185.301 is 194 ft.