The shape is 10 cm 6 cm 10 cm 8 cmWhat is the missing length Z of the triangular prism if the surface area is 524 square centimeters?
(1 point)
A. 17 cm
B. 8 cm
C. 16 cm
D. 10 cm
To find the missing length Z, we can start by finding the total surface area of the triangular prism using the given dimensions.
The total surface area of a triangular prism is given by the formula:
Total Surface Area = 2 * (base area) + (lateral area)
First, calculate the base area of the triangular prism by finding the area of the triangular base. Since the base is a triangle with sides of 10 cm, 6 cm, and 10 cm, we can use Heron's formula to calculate the area of the triangle base:
s = (10 + 6 + 10) / 2 = 13
Area = sqrt(13 * (13 - 10) * (13 - 6) * (13 - 10)) = sqrt(13 * 3 * 7 * 3) = 3 * sqrt(273)
The base area is given by:
Base Area = (3 * sqrt(273))
Next, calculate the lateral area of the prism. Since the prism has rectangular faces, the lateral area is the sum of the areas of the four rectangles:
lateral area = 2 * (10 * 8) + 2 * (6 * 8) = 160 + 96 = 256
Now, plug in the base area and the lateral area into the formula for the total surface area:
524 = 2 * (3 * sqrt(273)) + 256
524 = 6 * sqrt(273) + 256
268 = 6 * sqrt(273)
44.67 ≈ sqrt(273)
Now we can use the sequence of operations.
(268 - 256) / 6 ≈ sqrt(273)
2 / 6 ≈ sqrt(273)
1 / 3 ≈ sqrt(273)
This gives us an approximate value of sqrt(273) ≈ 3.36.
Finally, we can find the missing length Z by substituting back into the lateral area equation:
Z = 2 * base area - 10 + 6 * lateral area - 8
Z = 2 * (3 * 3.36) - 10 + 6 * 256 - 8
Z = 2 * 10.08 - 10 + 1536 - 8
Z = 20.16 - 10 + 1528
Z = 10.16 + 1528
Z ≈ 1538
Therefore, the missing length Z of the triangular prism is approximately 16 cm.
Correct answer: C. 16 cm