What is the minimum number of feet of fencing needed to create a rectangular garden with an area of 576ft?
Please help!!
LOL,square
min P, given A
A = L w so L = A/w
P =2L+2w
P = 2A/w + 2 w
2 w^2 - w P = -2 A
w^2 -(P/2)w = A
w^2 -P/2 w + P^2/16 = -A + P^2/16
(w-P/4)^2 = -(1/16)(16A-P^2)
vertex when w = P/4
(that is a square :)
Area = P^2/16 = (P/4)^2 of course
P given, max A
A = L w
P =2L+2w so L = (P-2w)/2
A = (P-2w)w/2
2A = -2w^2+Pw
d(2A)/dw = -4w +P
0(max or min) when P = 4 w
that is a square :)
Hi, Im sorry Im not understanding i keep getting 144 as my answer
sqrt 576 = 24
24*4 = 96
To calculate the minimum number of feet of fencing needed to create a rectangular garden with an area of 576ft, we need to find the perimeter of the rectangle.
The formula for the area of a rectangle is:
Area = length * width
In this case, the area is given as 576ft, so we need to find the length and width of the rectangle.
To find the length and width that maximize the area while having a fixed perimeter (minimum amount of fencing), we need to consider that a rectangle is symmetrical and has two equal sides.
Let's assume the length of the rectangle is L and the width is W.
So the perimeter would be:
Perimeter = 2 * (L + W)
To minimize the amount of fencing, we need to find the values of L and W that satisfy the equation:
2 * (L + W) = minimum
Now, let's solve for L and W.
Since the area of the rectangle is 576ft, we have:
L * W = 576
Since we know the relationship between L and W from above, we can substitute it into the equation and solve for L:
L * (576/L) = 576
By simplifying the equation, we get:
576 = L^2
Taking the square root of both sides:
L = sqrt(576)
L = 24
Now, substituting this value of L back into the equation to find W:
24 * W = 576
W = 576/24
W = 24
Thus, the length and width of the rectangle are both 24ft, which means the minimum number of feet of fencing needed would be:
Perimeter = 2 * (L + W)
Perimeter = 2 * (24 + 24)
Perimeter = 96ft
Therefore, the minimum number of feet of fencing needed to create a rectangular garden with an area of 576ft is 96ft.