Question

1) A box with a square base has a total surface area of 12m^2. Express the volume of the box as a function of the base edge.

2) A 10 foot tall lamppost causes a child 4 feet tall to cast a shadow of x feet on the ground. If the distance along the ground from the child to the lamppost is d meters, express x as a function.

Answer 1

let the side of the square base be x m
let the height be y m

I will assume the box has a lid so the surface area is made up of 2x^ + 4xy
then 2x^2 + 4xy = 12
4xy = 12-2x^2
y = (6-x^2)/(2x)

volume = x^2y
= x^2(6-x^2)/(2x)
= x(6-x^2)/2

for the second, you have similar triangles, so set up a ratio
4/x = 10/(x+d)
10x = 4x + 4d

take it from there