What is the shortest possible perimeter for an arrangement with an area of 15 square feet?
16 feet
What is the shortest possible perimeter for an arrangement with an area of 15 square feet
need help ASAP!!!
To find the shortest possible perimeter for an arrangement with an area of 15 square feet, we need to determine the shape that would minimize the perimeter. In this case, we're trying to find the shape with the smallest perimeter that still has an area of 15 square feet.
One way to approach this is to consider different shapes and see which one yields the smallest perimeter. Let's begin with a rectangle. The formula for the area of a rectangle is given by multiplying its length (L) by its width (W): Area = L * W. In this case, we know that the area is 15 square feet.
We can write an equation: 15 = L * W. To minimize the perimeter, we want to minimize the sum of the sides (perimeter) given the fixed area.
Using the equation for the area, we can express one variable in terms of the other. For example, L = 15 / W.
Substituting this expression for L in the formula for perimeter, which is given by P = 2L + 2W, we get: P = 2(15 / W) + 2W.
To find the minimum perimeter, we can take the derivative of the perimeter equation with respect to W, set the derivative equal to zero, and solve for W. This will give us the value of W that minimizes the perimeter:
dP/dW = -30/W^2 + 2 = 0.
Simplifying this equation, we get: -30/W^2 = -2.
By cross-multiplying, we have: -30 = -2W^2.
Dividing both sides by -2, we get: 15 = W^2.
Taking the square root of both sides, we find that W = sqrt(15).
Since we're seeking the length that minimizes the perimeter, we want to make L as close to the width W as possible. Therefore, L should also be approximately sqrt(15).
Now, we can calculate the shortest possible perimeter. Using the formula P = 2L + 2W, where L = sqrt(15) and W = sqrt(15), we substitute these values into the equation:
P = 2(sqrt(15)) + 2(sqrt(15)) = 4(sqrt(15)) ≈ 19.6 feet.
Hence, the shortest possible perimeter for an arrangement with an area of 15 square feet is approximately 19.6 feet.
the greatest area with the least perimeter is a circle
for regular polygons
... the more sides , the greater the ratio of area to perimeter