How Do You Minimize Cost While Building a Rectangular Enclosure?

[eownby77]

The manager of a department store wants to build a 600 square foot rectangular enclosure on the store's parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing at a cost of $7 per running foot. The fourth side will be built of cement blocks at a cost of $14 per running foot. Find the dimensions of the enclosure that will minimize the total cost of the building materials.

I started out with (2x)(2y) = 600 for the area. Solved for y to get y= 300/2x. What I don't get is that there are going to be two side lengths, and 3 of them will cost less than one. Would I maximize the dimensions to find the minimum costs? You can't just set it up with 3 sides being the same, because then it wouldn't be a rectangle.

The first derivative I think is 2 - 300/x^2. Critical points: d.n.e. at x=0, after setting 2-300/x^2, x= square root of 150

I don't know how to find the domain, or where I should go from there.

 

[Ray Vickson]

The manager of a department store wants to build a 600 square foot rectangular enclosure on the store's parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing at a cost of $7 per running foot. The fourth side will be built of cement blocks at a cost of $14 per running foot. Find the dimensions of the enclosure that will minimize the total cost of the building materials.

I started out with (2x)(2y) = 600 for the area. Solved for y to get y= 300/2x. What I don't get is that there are going to be two side lengths, and 3 of them will cost less than one. Would I maximize the dimensions to find the minimum costs? You can't just set it up with 3 sides being the same, because then it wouldn't be a rectangle.

The first derivative I think is 2 - 300/x^2. Critical points: d.n.e. at x=0, after setting 2-300/x^2, x= square root of 150

I don't know how to find the domain, or where I should go from there.

1) Define your variables, including units.
2) Write an expression for area in terms of the variables.
3) Write an expression for cost in terms of the variables.
4) Think about what you need to do to minimize something.

RGV