Suppose the area of a rectangle is 36 square inches. Find the Minimum perimeter for a rectangle with this area.
Please help asap. Thank you! (so much)
We know that A=l*w=36 and P=2(l+w)
From the area formula we see that l=36/w
Putting this into the perimeter formula we get
P=2(36/w + w) = 72/w +2w
dP/dw= -72/w^2 +2
Set that =0 and we get -144/w^2 +2=0
2 = 72/w^2, or w^2 = 36, thus w=6 and l=6 too.
I think you should be able to prove this for the general case too.
To find the minimum perimeter for a rectangle with an area of 36 square inches, we can use calculus to optimize the perimeter formula.
Step 1: Start with the area formula for a rectangle: A = length * width = 36 square inches.
Step 2: Substitute the expression for length into the perimeter formula: P = 2(length + width). Since we know that A = 36, we can solve for length in terms of width: length = 36/width.
Step 3: Plug in the expression for length into the perimeter formula: P = 2(36/width + width) = 72/width + 2width.
Step 4: Calculate the derivative of the perimeter function with respect to width to find the critical points: dP/dw = -72/width^2 + 2.
Step 5: Set the derivative equal to zero and solve for width: -72/width^2 + 2 = 0. Simplifying this equation, we get -144/width^2 + 2 = 0.
Step 6: Multiply both sides of the equation by width^2: 2 = 72/width^2.
Step 7: Cross-multiply to solve for width^2: width^2 = 36.
Step 8: Take the square root of both sides to find the width: width = 6.
Step 9: Since the width and length have the same value in this case, the length is also 6.
Step 10: Plug the values of width and length back into the perimeter formula to find the minimum perimeter: P = 2(6 + 6) = 24 inches.
Therefore, the minimum perimeter for a rectangle with an area of 36 square inches is 24 inches, with a width and length of 6 inches each.