A pedestal in a craft store is in the shape
of a triangular prism. The bases are right triangles with side lengths of
12Â centimeters, 16 centimeters, and 20 centimeters. The store owner
wraps a piece of rectangular cloth around the pedestal, but does not
cover the identical bases of the pedestal with cloth. The area of the cloth
is 192 square centimeters.
a. What is the distance around the base of the pedestal? How do
you know?
b. What is the height of the pedestal? How did you find your answer?
a. To find the distance around the base of the pedestal, you need to calculate the perimeter of one of the triangular bases. The base is a right triangle with side lengths of 12 centimeters, 16 centimeters, and 20 centimeters.
To calculate the perimeter of a triangle, you add up the lengths of all three sides. In this case, the perimeter of the triangle can be found by adding 12 + 16 + 20 = 48 centimeters. So, the distance around the base of the pedestal is 48 centimeters.
b. To find the height of the pedestal, you can use the formula for the surface area of a triangular prism. The surface area is equal to the sum of the areas of the two triangular bases, plus the area of the three rectangular faces.
In this case, the surface area is given as 192 square centimeters. Since the triangular bases are identical, the sum of their areas is half of the total surface area, which is 192/2 = 96 square centimeters.
The area of a right triangle can be calculated using the formula: (base * height) / 2.
Plugging in the known side lengths, we have:
96 = (12 * height) /2 + (16 * height) /2
Simplifying the equation, we have:
96 = (12 * height + 16 * height) / 2
Multiplying both sides of the equation by 2 to eliminate the denominator, we get:
192 = 12 * height + 16 * height
Combining like terms, we have:
192 = 28 * height
Dividing both sides of the equation by 28, we find:
height = 192 / 28
Evaluating the division, we find that the height of the pedestal is approximately 6.857 centimeters.
a. you know the right triangle sides, add them
b. surface area=base perimeter*height=192, solve for height.