A construction company wants to build a rectangular enclosure with an area of 1000 square feet by fencing in three sides and using its office building as the fourth side. Your objective as supervising engineer is to design the enclosure so that it uses the least amount of fencing. Proceed as follows. (a) Let x and y be the dimensions of the enclosure, where x is measured parallel to the building, and let L be the length of fencing required for those dimensions. Since the area must be 1000 square feet, we must have xy = 1000. Find a formula for L in terms of x and y, and then express L in terms of x alone by using the area equation. (b) Are there any restrictions on the value of x? Explain.

(a) L = x + 2y = x + 2000/x

(b) 0 < x < L, for obvious reasons

For optimum (minimum) L, dL/dx = 0

1 - 2000/x^2 = 0
x = sqrt2000 = 44.74 feet
y = 1000/x = 22.37 feet

(a)

Let's proceed with step (a) as follows:

1. Let x and y be the dimensions of the enclosure, where x is measured parallel to the building.

2. The area of the enclosure is given as 1000 square feet, so we have xy = 1000.

3. The length of fencing required for the three sides is given by L = x + 2y, since x is parallel to the building and two sides of length y are perpendicular to x.

4. Express L in terms of x and y: L = x + 2y.

5. Substitute xy = 1000 into the equation for L: L = x + 2(1000/x).

So, the formula for L in terms of x and y is L = x + 2(1000/x).

(b)
There are no explicit restrictions on the value of x given in the problem. However, we may need to consider practical restrictions based on the actual dimensions of the building and available space. Additionally, x and y should both be positive since we are dealing with lengths in this problem.

(a) To find a formula for L in terms of x and y, we need to consider the perimeter of the enclosure. Since there are three sides that need to be fenced in, the perimeter will be the sum of the lengths of these three sides.

The length of the side parallel to the building is x, and the two other sides have a combined length of y. Therefore, the perimeter L can be expressed as:

L = x + y + y

Simplifying this equation, we get:

L = x + 2y

Now, we can express y in terms of x using the area equation. We know that xy = 1000, so we can solve for y:

y = 1000/x

Substituting this value for y in the formula for L, we get:

L = x + 2(1000/x)

(b) Yes, there are restrictions on the value of x. Since the dimensions of the enclosure must be positive (as we can't have negative lengths), x and y must both be positive values.

In addition, to minimize the amount of fencing required, we want to minimize L. To do this, we can find the minimum value of L by finding the critical points of the function L(x).

Differentiating L(x) with respect to x, we get:

L'(x) = 1 - 2000/x^2

Setting L'(x) equal to zero and solving for x, we get:

1 - 2000/x^2 = 0
x^2 = 2000
x = sqrt(2000) = 44.721

So, the minimum value of L occurs when x is approximately 44.721. Therefore, there is a restriction that x must be greater than zero and should ideally be close to 44.721 to minimize the amount of fencing required.