Use the image to answer the question.
An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side of the triangle measures 10 centimeters. The right side of the triangle also measures 10 centimeters. The perpendicular height from the top vertex of the triangle to the base measures 6 centimeters. The base of the triangle measures 8 centimeters.
What is the missing length l
of the triangular prism if the surface area is 524 square centimeters?
(1 point)
Responses
17 cm
17 cm
16 cm
16 cm
8 cm
8 cm
10 cm
10 cm
The surface area of a triangular prism can be calculated using the formula S = 2B + Ph, where B is the area of the triangular base, P is the perimeter of the base, and h is the height of the prism.
Given:
Base of the triangle = 8 cm
Left side of the triangle = 10 cm
Right side of the triangle = 10 cm
Height of the triangle = 6 cm
First, calculate the area of the triangular base:
Area of the triangular base = (1/2) * base * height
Area of the triangular base = (1/2) * 8 * 6
Area of the triangular base = 24 square centimeters
Next, calculate the perimeter of the triangular base:
Perimeter of the triangular base = Left side + Right side + Base
Perimeter of the triangular base = 10 + 10 + 8
Perimeter of the triangular base = 28 cm
Now, calculate the surface area of the prism using the formula S = 2B + Ph:
Surface area = 2 * 24 + 28 * 6
Surface area = 48 + 168
Surface area = 216 square centimeters
Since the given surface area is 524 square centimeters, and we have only accounted for 216 square centimeters, the missing length must be added to reach the total surface area of 524 square centimeters. The total missing length required to reach 524 square centimeters is 524 - 216 = 308 square centimeters.
Let's try a length of 17 cm as a guess, since it is not already listed as an option.
Adding 17 cm as the missing length would include the length of the right triangular side.
Ph = 4∗17 = 68 cm
Surface area would be Surface area = 2 * 24 + 68
Surface area = 48 + 68
Surface area = 116 square centimeters
This would still be below the required surface area of 308 square centimeters, so the correct missing length must be higher.
Let's try a length of 16 cm next as it is not listed as an option.
Adding 16 cm as the missing length would include the right-triangular side length.
Ph = 4∗16 = 64 cm
Surface area would be Surface area = 2 * 24 + 64
Surface area = 48 + 64
Surface area = 112 square centimeters
Again, this is below the required surface area of 308 square centimeters, so the correct accurate length must be lower.
This process of elimination reveals that the correct missing length l for the triangular prism is 10 cm, as this will fulfill the requirement to reach the surface area of 524 square centimeters as per calculations.