calculus, dimensions

Question

A fence is to be built to enclose a rectangular area of 280 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Dimensions?

Answer 1

Since the area of a rectangle is length time width we have the relationship 280 = xy. At this point we do not know if x is length or width, but it should not matter.

We need to develop a cost function in terms of either x or y, find the derivative of that function, set it equal to zero and determine "critical" numbers. Finally we should probably show that the critical numbers we discover minimize the cost.

For the perimeter (length we have 2x + 2y. 3 of the sides cost $3 and the 4th side $16 foot.

3 (3x) + 16 y = cost. But this function depends on both x and y. We know for example that y = 280/x, so substitute.

Whoops, original cost function not correct. Should be C = 3y + 16y + 3(2x) = 19y + 6x. So,

C = 6x + 19 (280x^-1)

See if you can find dC/dx, set equal to zero, etc.

Answer 2

The problem asks us to minimize the cost of the fence. In calculus, the first derivative allows us to find relative mins or maxes, so we will need the first derivative of some cost equation to find the minimum.

Let x = the length of one side of the rectangular area

Let y = the length of a side adjacent to side x

Then A = the rectangular area = xy = 280 sq ft

Let C = the cost of the fencing material = ($3/ft)(x ft) + ($3/ft)(y ft) + ($3/ft)(x ft) + ($16/ft)(y ft)

Simplified, C = 6x + 19y

We can't find C' yet, because C is really C(x,y) and we want C(x), so that C is written in terms of only one variable. So we have to use the area relationship and solve for y:

Since xy = 280, y = 280/x

Then, C = 6x +19y = 6x + 19(280/x) = 6x + 5320/x = 6x + 5320x^(-1)

Now, using the power rule, we can take the derivative of C with respect to x:

C' = 6 - 5320x^(-2) = 0 ==> 6 = 5320x^(-2) ==> x^2 = 5320/6 ==> x = sqrt(5320/6)

x = 29.78 ft

Then y = 280/x, so y = 9.4 ft