Question

There are ​$528 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs ​$9 per​ foot, and the fencing for the other three sides costs ​$3 per foot. The picture on the right depicts this situation. Consider the problem of finding the dimensions of the largest possible garden.

Answer 1

Answer:

  22 ft by 44 ft, with 22 ft parallel to the road

Step-by-step explanation:

Problems in optimizing rectangular area for a given perimeter or perimeter cost all have a similar solution: the length (or cost) of one pair of opposite sides is equal to that of the other pair of opposite sides.

Here, that means that the sides perpendicular to the road will have a total cost of $528/2 = $264, so will have a total length of $264/($3/ft) = 88 ft. Since it is a rectangle, the dimension perpendicular to the road is 44 ft.

Likewise, the sides parallel to the road will have a total cost of $264. If x is the length in that direction, this means ...

  9x +3x = 264

  12x = 264

  264/12 = x = 22

The length of the garden parallel to the road is 22 ft.

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If you solve this directly, you get the same result. Let x be the distance parallel to the road. Then the cost of fence for the two sides parallel to the road is (3x +9x) = 12x.

The length of fence perpendicular to the road will use the remaining cost, so that length will be (528 -12x)/(2·3). (Half of the remaining fence is used on each of the two parallel sides.) This expression for length simplifies to (88-2x).

Then the area of the garden will be the product of its length and width:

  area = x(88 -2x)

This is the equation for a downward-opening parabola with zeros at x=0 and x=44. The vertex is located halfway between those zeros, at x = 22.

The dimensions of the largest garden are 22 ft parallel to the road and 44 ft perpendicular to the road.