Question

Concrete planter is formed from a square-based pyramid that was inverted and placed

inside a cube.
A.
What is the slant height of the pyramid?
2 yd
2 yd
What is the surface area of the composite figure?
HINT: The surface area consists of lateral faces of the
inside of the inverted pyramid and the remaining 5
faces of the cube.​

Answer 1

The slant height of the pyramid refers to the height of one face-triangle. Now, the cube has dimensions of 2 x 2 x 2, which means the height, base and length of the pyramid are equal, that's why is a squared pyramid.

Notice that the slant height forms a right triangle with two faces of the cube, and the pyramid intercepts the cube at the middlepoint of the plane.

First, we need to find the half-length of a diagonal, which is a leg of the right triangle formed by the slant height.

`d^(2)=2^(2) +2^(2)\\ d=√(4+4)= √(8)\\ d=2√(2)`

The part related with the slant height is

`d_(half) =(2√(2) )/(2)=√(2)`

So, the slant height is

`h_(slant)= \sqrt{(√(2) )^(2) +2^(2) }=√(2+4)=√(6)`

Therefore, the slant height is the square root of 6 yards.

The surface area of a squared pyramid is

`S_(area)=2(2)(√(6))+(2)^(2) =4√(6)+4 \approx 13.8yd^(2)`

The five faces of the cube have a surface area

`S_(cube)=5(2)^(2) =20yd^(2)`

Therefore, the composite surface area of the figure is

`S_(total)=13.8+20=33.8yd^(2)`