Find the dimensions of the rectangle with area 256 square inches that has minimum perimeter, and then find the minimum perimeter.
Dimensions:
Minimum Perimeter:
dimensions
x:16 y: 16
perimeter: 64
256/16 =16 🙄
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Dimensions: It's the ghost of a rectangle – it's invisible! 😄
Minimum Perimeter: Just don't draw the rectangle, and the perimeter will be zero! Easy peasy, lemon squeezy! 🍋😉
To find the dimensions of the rectangle with the minimum perimeter, we need to use the given information that the area of the rectangle is 256 square inches.
Let's assume the length of the rectangle is x inches. Then the width of the rectangle would be the area divided by the length: width = 256 / x inches.
The formula for the perimeter of a rectangle is P = 2(length + width). Substituting the values, we get P = 2(x + (256/x)) inches.
To find the minimum perimeter, we can differentiate the equation with respect to x and set it to zero to find the critical points.
dP/dx = 2 - (512/x^2) = 0
Solving this equation, we get x^2 = 512 / 2, which simplifies to x = √256 = 16 inches.
Thus, the length of the rectangle is 16 inches, and the width is 256 / 16 = 16 inches as well.
Therefore, the dimensions of the rectangle with the minimum perimeter are 16 inches by 16 inches.
To find the minimum perimeter, substitute the dimensions into the perimeter formula: P = 2(16 + 16) inches = 64 inches.
So, the minimum perimeter of the rectangle is 64 inches.
length -- x
width -- y
xy = 256
y = 256/x
perimeter = 2x + 2y
= 2x + 2(256/x) = 2x + 512/x
d(perimeter)/dx = 2 - 512/x^2
= 0 for a min of perimeter
2 = 512/x^2
x^2 = 256
x = √256 = 16
if x = 16, then y = 256/16 = 15
perimeter = 2(16)=2(16) = 64