A rectangular field is to be enclosed on four sides with a fence. Fencing costs $4 per foot for two opposite sides, and $8 per foot for the other two sides. Find the dimensions of the field of area 880 ft 2 that would be the cheapest to enclose.

the length  41.95 feet and the width 20.98 feet

Step-by-step explanation:

We know that "x" and "y" would be the sides of the rectangle.

Therefore we have that the area would be equal to:

x * y = 880

if we solve y, we have to:

y = 880 / x

Now, the cost would be the multiplication of the price by the length of the sides (each side is put twice, since the price goes by the perimeter of the figure):

c = 4 * 2x + 8 * 2y

replacing:

c = 8x + 16 * 880 / x

Now, if we derive with respect to x we are left with:

c '= 8 - 14080 / x ^ 2

if we derive again:

c '' = 28160 / x ^ 3

we calculate the critical point that is when c '= 0

0 = 8 + 14080 / x ^ 2

x ^ 2 = 14080/8

x ^ 2 = 1760

x = + -41.95

Now for x = 41.95, we evaluate in the second derivative to know if it is a minimum or maximum value:

28160 / 41.95 ^ 3 = 0.381, is a number greater than 0, therefore it is a minimum value, therefore it is value is correct, now for y:

y = 880 / 41.95 = 20.98

which means that for it to be cheaper the length must be 41.95 feet and the width 20.98 feet

step-by-step explanation: