Christian collects model cars and planes. He has a
display showcase of all of his collectors' items. Solve
this real-world problem to find the surface area of the
display showcase if it is the shape of a triangular prism
with the following dimensions: =9 inches, =17
inches, =10 inches, =12 inches, = 17 and =10
inches.
To find the surface area of the triangular prism display showcase, we first need to calculate the area of each of the five faces and then add them together.
1. The two triangular faces:
Each triangular face can be divided into a right-angled triangle and a smaller triangle (from the center of the base to the top). We can find the area of each right-angled triangle using the formula: 0.5 * base * height.
Area of the right-angled triangle = 0.5 * 9 * 17 = 76.5 square inches
There are two right-angled triangles, so the total area for both is 2 * 76.5 = 153 square inches.
For the smaller triangle, we can use the Pythagorean theorem to find the height.
h^2 + 8.5^2 = 17^2
h^2 + 72.25 = 289
h^2 = 216.75
h ≈ 14.72 inches
Now, we can find the area of the smaller triangle:
Area of the smaller triangle = 0.5 * 9 * 14.72 ≈ 66.24 square inches
Since there are two smaller triangles, the total area for both is 2 * 66.24 ≈ 132.48 square inches.
The total area for the two triangular faces is 153 + 132.48 ≈ 285.48 square inches.
2. The three rectangular faces:
The total area of the three rectangular faces can be found by calculating the area of each rectangle individually and then adding them together.
Area of rectangular face = length * width
Area of the first rectangular face = 10 * 17 = 170 square inches
Area of the second rectangular face = 10 * 12 = 120 square inches
Area of the third rectangular face = 9 * 17 = 153 square inches
The total area for the three rectangular faces is 170 + 120 + 153 = 443 square inches.
Therefore, the surface area of the triangular prism display showcase is 285.48 + 443 = 728.48 square inches.