Graham carves a block of wood shaped like a cube with a square pyramid on top. How much paint will Graham need to paint the toy?
To calculate the surface area of the cube and square pyramid, we need to find the area of each face and add them together:
1. Surface area of the cube:
- The cube has 6 faces, all of which are squares.
- If the length of each side of the cube is "s," then the area of each face is s^2.
- So, the total surface area of the cube is 6s^2.
2. Surface area of the square pyramid:
- The square pyramid has 5 faces - 1 square base and 4 triangular faces.
- If the base of the square pyramid has side length "s," then the area of the base is s^2.
- The area of each triangular face can be calculated using the formula for the area of a triangle: 0.5 * base * height.
- The height of the pyramid can be calculated using the Pythagorean theorem as h = sqrt(s^2 + (0.5s)^2) = sqrt(5/4) * s.
- So, the area of each triangular face is 0.5 * s * sqrt(5/4) * s = 0.5 * sqrt(5) * s^2.
- Therefore, the total surface area of the square pyramid is s^2 + 4 * (0.5 * sqrt(5) * s^2).
Adding the surface areas of the cube and square pyramid together:
Total surface area = 6s^2 + s^2 + 4 * (0.5 * sqrt(5) * s^2)
= 7s^2 + 2 * sqrt(5) * s^2
= (7 + 2 * sqrt(5)) * s^2
Now, if each side of the cube has a length of 10 cm (for example):
Total surface area = (7 + 2 * sqrt(5)) * 10^2
≈ (7 + 2 * 2.236) * 100
≈ 11.472 * 100
≈ 1147.2 cm^2
Given the surface area of the toy, we would need to calculate the amount of paint required for each cm^2 and then multiply it by the total surface area.