A triangular prism has a surface area of 122.9 in², and its known
dimensions are as follows: a base of 3 inches, a slant length of 4 inches, and a length of 10 inches. Determine the height of the height of the triangular face, round to the tenth (one-decimal place).
The surface area of a triangular prism can be calculated by adding the areas of the two triangular faces and the three rectangular faces.
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2
Let's first calculate the area of a triangular face:
Area of triangular face = (3 * height) / 2
We are given that the slant length is 4 inches, so we can use the Pythagorean theorem to find the height of the triangular face:
4^2 = 3^2 + height^2
16 = 9 + height^2
height^2 = 16 - 9
height^2 = 7
height = sqrt(7)
Now we can substitute the value of height in the formula to calculate the area of the triangular face:
Area of triangular face = (3 * sqrt(7)) / 2
Since there are two triangular faces, the total area of the triangular faces is:
2 * [(3 * sqrt(7)) / 2] = 3 * sqrt(7)
The surface area is given as 122.9 in², which equals the sum of the areas of the triangular faces and the rectangular faces:
3 * sqrt(7) + 3 * (base * length)
We are given that the length is 10 inches and the base is 3 inches. Substituting these values, we get:
3 * sqrt(7) + 3 * (3 * 10) = 122.9
Simplifying the equation:
3 * sqrt(7) + 90 = 122.9
3 * sqrt(7) = 122.9 - 90
3 * sqrt(7) = 32.9
Dividing both sides by 3:
sqrt(7) = 32.9 / 3
sqrt(7) ≈ 10.97
Rounding to one decimal place:
Height of the triangular face ≈ 10.97 inches