SOLUTION: A prism has congruent isosceles triangle for 2 of its faces. The ratio of one of the equal sides of one triangle to its base to the long edge of the prism is 5:8:9. The total volum
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Question 997512: A prism has congruent isosceles triangle for 2 of its faces. The ratio of one of the equal sides of one triangle to its base to the long edge of the prism is 5:8:9. The total volume of the solid is 6,912 cubic centimeter.
1. What is the perimeter of one triangle face?
2. What is the surface area of the entire solid? Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! equal legs of the isosceles triangle = 5x
base of the isosceles triangle = 8x
height of the isosceles triangle becomes 3x.
this is derived by dropping a perpendicular from the vertex of the isosceles triangle to its base which splits the base into 2 equal segments of 4x each.
solve for the height of the isosceles triangle by using one of the right triangles formed and pythagorus formula to get:
(4x)^2 + h^2 = (5x)^2 which becomes:
16x^2 + h^2 = 25x^2
solve for h^2 to get h^2 = 9x^2
solve for h to get h = 3x.
now you can find the area of each isosceles triangle.
area = 1/2 * b * h which becomes:
area = 1/2 * 8x * 3x which becomes:
area = 12x^2.
the volume of the prism is equal to the area of the base * the height.
the height of the prism is 9x.
volume is equal to 12x^2 * 9x = 108x^3.
since the volume of the prism is given as 6912, the formula becomes:
6912 = 108x^3.
solve for x to get x = 4.
the perimeter of one triangle face is equal to 5x + 8x + 5x = 18x.
x = 4, so the perimeter of one triangle face is equal to 18 * 4 = 72 centimeters.
the surface area of the prism is the sum of the area of both triangular faces plus the sum of the area of the 3 rectangular faces.
each triangular face has an area of 1/2 * 3x * 8x = 1/2 * 3 * 4 * 8 * 4 = 1/2 * 12 * 32 = 6 * 32 = 192 square centimeters.
two triangular faces have a total surface area of 192 * 2 = 384 square centimeters.
2 of the rectangular faces have an area of 9x * 5x each = 45x^2 45 * 4^2 = 45*16 = 720 square centimeters each * 2 = 1440 square centimeters.
the third rectangular face has an area of 9x * 8x = 72x^2 = 72 * 4^2 = 72 * 16 = 1152 square centimeters.
total surface area = 384 + 1440 + 1152 = 2976 square centimeters.
your solution is:
the perimeter of one triangular face = 72 centimeters.
the surface area of the prism = 2976 square centimeters.
here's a diagram that might help you to understand.
