area/volume geometry

Question

A nylon tent is made of a pyramid on top of a cube. If the front of the tent is 14 feet and the overall height is 20 feet how much nylon is required to make the tent?

Answer 1

We need to add the surface area of the cube to the surface area of the pyramid.

The front of the tent is 14 feet, which means each side of the cube is 14 feet.

The surface area of the cube would be the sum of the areas of the six identical squares that make up the cube (the area of one square being 14 x 14).

If the overall height is 20 feet, that means the height of the pyramid is 6 feet. Also, since it is on top of the cube, it is a pyramid with a square base with sides equal to 14 feet. The surface area of a pyramid is the area of the base + 1/2 * slant height * base perimeter. The area of the base is the area of a square with side 14 feet and the base perimeter is the perimeter of a square with side 14 feet. The slant height is going to be the hypotenuse of a right triangle whose legs are the height of the pyramid and one half the width (7 feet).

Answer 2

Looking at one of the four sides of the pyramid, you have a triangle. Looking at it from the side, you see that you have a triangle with height (h) = 20 and a base (b) = 7. So the length of the midpoint running up the side triangle (in the middle of the triangle) is h² = 20² + 7². Solving, we have h = 21.19 ft.

So, the area of one of the triangle sides is (1/2)bh = .5(14)(21.19) = 148.33 ft²

Now then, all 4 of the triangle areas = 4(148.33) = 593.31 ft2

If you consider the base as being part of the tent, you have to add Area (base) = 14² = 196 ft²