Question

A 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?

B. What is the surface area of the composite figure?

C. How many cubic yards of concrete are needed to make the planter?

Answer 1

A. 2.24 yd

B. 53.67 yd²

C. 5.33 yd³

Explanation:

Here we have;

Height of the pyramid = 2 yd

Side length of pyramid base = 2 yd

Therefore, the slant height forms a right triangle with the height of the pyramid and half the base hence;

(Slant height)² = (2·yd)² + (2·yd/2)² = 4·(yd)² + (yd)² = 5·(yd)²

∴ Slant height = √5 yards = 2.24 yd

B. The surface area of the cube with one side open is found as follows;

Surface area of container cube = 5 × (2·yd)² = 20 yards²

The surface area of the pyramid = Base area + 1/2 perimeter of base × Slant height

Since the base is open, we have;

The surface area of the pyramid = 1/2 perimeter of base × Slant height

= 1/2 × (4 × (2·yd))×yd·√5 = 4·yd×yd·√5 = 4·√5·(yd)²

Hence total surface area of the planter = Surface area of container cube + surface area of the pyramid

total surface area of the planter = 20·yd² + 4·√5 yd² = 24·√5 yards²

C. The volume of concrete needed to make the planter is the volume of the cube concrete container less the volume of the inverted pyramid

Volume of the cube = 2 × 2 × 2 = 8 yd³

Volume of the inverted pyramid = 1/3 × Base area × Height

Volume of the inverted pyramid = 1/3 × 2 × 2 × 2 = 8/3 yd³

Therefore, volume of the concrete needed = 8 yd³ - 8/3 yd³ = 16/3 yd³ = 5.33 yd³.

Answer 2

Notice that each edge of the cube is 2 yards long, and the height of the pyramid is 2 yards long too.

The slant height refers to the height of a triangle on its face, which forms a right triangle with the height of the pyramid and half side. Using Pythagorean's Theorem, we have

`s^(2)=1^(2) +2^(2)\\ s=√(5)`

Therefore, the slant height is the square root of 5, in yards units. (A)

The surface of the composite figure is the sum of the surface area of both volumes.

`S_(composite)=S_(cube) +S_(pyramid)\\ S_(composite)=5(2)^(2) +(2)^(2) +(1)/(2) (8)(√(5) )=20+4+4√(5) \\ S_(composite)\approx 32.9 yd^(2)`

Therefore, the surface of the composite figure is 32.9 square yards. (B)

The concrete needed will fill the empty space between the cube and the pyramid, so we have to find the difference between their volumes.

`V_(concrete)=V_(cube) -V_(pyramid)=2^(3)-(1)/(3)(2)^(2) (2)=8-(8)/(3) \\V_(concrete) \approx 5.33 yd^(3)`

Therefore, we need 5.33 cubic yards of concrete to make the planter. (C)