a hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence. if the two sides that come out from the store front cost $3 per running foot and the side parallel to the store from costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost of the fence.

Question

a hardware store owner chooses to enclose an 800 square foot rectangular area in front of her store so that one of the sides of the store will be used as one of the four sides of the fence. if the two sides that come out from the store front cost $3 per running foot and the side parallel to the store from costs $5 per running foot for materials, then find the dimensions of the fence that will minimize the cost of the fence.

Answer 1

The coordinate of all relative maxima 2x^4+x^3-33x^2-16x+16

Answer 2

To minimize the cost of the fence, we need to find the dimensions of the rectangular area that will use the least amount of expensive materials (the $5 per running foot side) while still enclose an area of 800 square feet.

Let's denote the width of the rectangular area as "x". Since one side of the store will be used as one of the sides of the fence, the other side of the fence will also have a width of "x". The length of the rectangular area will be denoted as "y".

To calculate the dimensions, we need to set up an equation based on the given information:
1. The area of a rectangle is given by the formula: Area = length × width.
2. The length of the rectangular area will be the sum of the lengths of the two sides coming out from the store, which are both "y".
3. The width of the rectangular area will be the sum of the width of one side coming out from the store and the width of the side parallel to the store, which are both "x".

Based on these considerations, we have the equation:
800 = (2y + x) × x

Now, let's solve for "x" in terms of "y":
800 = (2y + x) × x
800 = 2xy + x^2
x^2 + 2xy - 800 = 0

Next, we need to minimize the cost of the fence. The cost will be the sum of the costs of the two sides that come out from the store front (which cost $3 per running foot), and the side parallel to the store front (which costs $5 per running foot). The total cost can be represented as:
Cost = 3(2y) + 3x + 5x
Cost = 6y + 8x

Now, we have a relationship between the cost and the dimensions of the fence. We can substitute the value of "x" from the area equation into the cost equation, to get the cost in terms of "y" only:
Cost = 6y + 8x
Cost = 6y + 8(800/(2y + x))

Simplifying this equation will yield a cost function in terms of "y" only, which we can then minimize to find the dimensions that minimize the cost.