Question

i would like to create a rectangular vegetable patch. the fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. i have a budget of $192 for the project. let x be the length in feet of the east side and of the west side. find the function for area in square feet a, in terms of x only, subject to any constraints.

Answer 1

The area A of a rectangular vegetable patch in terms of the length x of the east and west sides, given the budget constraint and cost per foot of fencing, is A(x) = x(48 - 2x), with x constrained between 0 and 24 feet.

Step-by-step explanation:

To find the function for area A of a rectangular vegetable patch in terms of the length x of the east and west sides, we need to consider the budget for the fencing and the cost per foot of fencing the north and south sides. Given that the fencing for the east and west sides is $4 per foot, the overall cost for fencing these two sides would be $4x + $4x = $8x.

Similarly, if y represents the length of the north and south sides, the cost for fencing these sides would be $2y + $2y = $4y. With a total budget of $192, we can formulate the budget constraint as $8x + $4y = $192.

To express the area A as a function of x only, we need to solve for y using the budget constraint. Rearranging the equation we get y = ($192 - $8x)/$4. Since the area of a rectangle is A = x×y, we can substitute y with the equation derived from the budget constraint to get A(x) = x × (($192 - $8x)/$4). Therefore, the area function in terms of x is A(x) = x × (48 - 2x). The constraint on x is that it must be a positive number and it should fulfill the budget restriction, meaning $8x ≤ $192, hence, 0 < x ≤ 24.