What is the surface area of a triangular prism shaped doorstop with a height of 1.5 inches, a width of 1.5 inches, a length of 5 inches, and a hypotenuse of 5.22 inches?
A triangular prism has two congruent triangular bases and three rectangular faces.
The base triangle has a height of 1.5 inches and a base of 1.5 inches, so its area is:
(1/2) x 1.5 inches x 1.5 inches = 1.125 square inches
The rectangular faces have dimensions of 1.5 inches x 5 inches, so each has an area of:
1.5 inches x 5 inches = 7.5 square inches
There are three rectangular faces, so their total area is:
3 x 7.5 square inches = 22.5 square inches
To calculate the total surface area, we need to add the area of the two triangular bases. To find the area of the base triangle, we can use its hypotenuse and height:
Base = (1/2) x Base x Height
We know that the hypotenuse is 5.22 inches, and the height is 1.5 inches. To find the base, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
where a and b are the legs of the triangle, and c is the hypotenuse.
Rearranging the formula, we get:
a^2 = c^2 - b^2
a^2 = 5.22^2 - 1.5^2
a^2 = 25.9284 - 2.25
a^2 = 23.6784
a = 4.866 inches (rounded to three decimal places)
Now we can calculate the area of each base triangle:
Base = (1/2) x 1.5 inches x 4.866 inches = 3.6685 square inches
The total surface area is:
2 x 3.6685 square inches + 22.5 square inches = 29.837 square inches
Therefore, the surface area of the triangular prism doorstop is 29.837 square inches.