Question in the image

Question

Describe the effect of each change of Linear Dimensions on the Surface Area:

1. The dimensions of a rectangular prism are doubled.
2. The dimensions of a square pyramid are divided by 3.

Answer 1

1) use the formula for the surface area of a rectangular prism:A=2(wl+hl+hw)

1. double all dimensions: A = 2*(2w*2l + 2h*2l + 2h*2w) = 2*(2l+hl+hw)*4 --> the surface area is quadrupled

2) use the formula for the surface area of a square pyramid A=a²+2a*sqrt(a²/4+h²)

1. divide all dimensions by 3: A = (a/3)² + 2*(a/3)*sqrt(a²/(9*4)+h²/9)
2. expand: 1/9*a²+2a/3*sqrt((a²/4+h²)/9)
3. pull the 1/9 out of your square root: 1/9*a²+2*a/3*1/3*sqrt(a²/4+h²)
4. expand the second term: 1/9*a²+1/9*2*a*sqrt(a²/4+h²)
5. pull the 1/9 out: 1/9*(a²+2*a*sqrt(a²/4+h²))
6. The square pyramids's surface area is multiplied by a factor of 1/9