Determine minimum cost of container - calculus question see below for information

Question

An open topped square based prisim(ex. box) is to be used by student council for selling popcorn. The container is made out of a card stock and costs $0.000087/cm²

If there is to be a total of 1250cm³ of popcorn sold then determine the minimum cost of the container.

Answer 1

Need to find the minimum surface area of the topless box then multiply by the cost/cm².

Let x be the dimension of the square base and y be the height.

Volume, V = x²y = 1250 cm³

y = 1250/x²

Surface Area, S= bottom area + 4 (area of a side) = x²+xy=x²+5000/x

Take the derivative of S(x), set to zero and solve for x.

S'(x)=2x-5000/x²

2x-5000/x²=0

2x³=5000

x = 2500^1/3

S(2500^1/3) = 2500^2/3+5000/2500^1/3 = (2500+5000)/2500^1/3 = 7500/2500^1/3 ≈552.6

Cost = 0.000087(552.6) ≈ $0.048 ≈ $0.05 = 5 cents

Answer 2

Hi Mikayla!

Start by drawing a picture of the open-topped square-based prism. Let the length of the sides of the square on the bottom of the box be x, and the height of the box be y.

We know that

V=1250cm^3 = (x^2)(y)

Solving for y

y=1250/(x^2)

We also know

SA = x^2 + 4xy

Substituting in for y

SA = x^2 + 5000/x

We know that x cannot be zero, and x must be positive, so we are looking for a positive critical point in the interval (0, infinity), which we can obtain by looking for critical points.

Taking the derivative of SA, we get

SA' = 2x - 5000/(x^2)

Solving this gives us x=13.57

Plugging back in the original Volume equation gives us y =6.79

Substituting these values into the SA equation, we get

SA = (13.57) ^2 + 4(13.57) (6.79) = 552.7061 cm^2

This is the amount of card stock needed, so multiplying this by the given cost, 0.000087 / cm^2 gives us a final answer of $0.0489