How would you set up this problem if instead of using 1000 feet of fencing, only 750 feet of fencing is used? Solve the problem to find the maximum area..
To solve this problem, we need to determine the dimensions that would maximize the area of the rectangular enclosure using only 750 feet of fencing.
Let's assume the rectangular enclosure has a length of L and a width of W. We can express the perimeter of the enclosure using the given fencing as:
Perimeter = 2L + 2W
Since we are given that the total length of the fencing is 750 feet, we can write the equation using the given perimeter:
2L + 2W = 750
To find the maximum area, we need to maximize the product of the length and width. The area of the enclosure is given by:
Area = L * W
To solve this problem, we can express one variable in terms of the other using the perimeter equation, substitute it into the area equation, and then find the maximum value of the area.
Let's solve the perimeter equation for L:
2L + 2W = 750
2L = 750 - 2W
L = (750 - 2W)/2
L = 375 - W
Now, substitute this expression for L in the area equation:
Area = L * W
Area = (375 - W) * W
Area = 375W - W^2
To find the maximum area, we can take the derivative of the area function with respect to W and set it to zero, then solve for W.
d(Area)/dW = 375 - 2W
Setting this equation equal to zero:
375 - 2W = 0
2W = 375
W = 375/2
W = 187.5
Now substitute the value of W back into the equation for L:
L = 375 - W
L = 375 - 187.5
L = 187.5
So, when 750 feet of fencing is used, the dimensions that maximize the area of the rectangular enclosure are a length of 187.5 feet and a width of 187.5 feet. To find the maximum area, we substitute these values into the area equation:
Area = L * W
Area = 187.5 * 187.5
Area = 35,156.25 square feet
Therefore, the maximum area of the rectangular enclosure using 750 feet of fencing is 35,156.25 square feet.