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.
