An architect needs to design a rectangular room with an area of
75 ft
What dimensions should he use in order to minimize the perimeter? Round to the
nearest hundredth, if necessary
a square has maximum area for a given perimeter. Or, it has minimum perimeter for a given area. So, if each side is √75 = 8.66 ft that will minimize the perimeter.
To minimize the perimeter of a rectangular room with an area of 75 ft, you need to find the dimensions that will give you the smallest perimeter.
Let's assume the dimensions of the room are length (L) and width (W). The perimeter (P) is given by the formula:
P = 2L + 2W
Since the area (A) of the room is given as 75 ft, we have:
A = L * W
Now, we can solve for either L or W in terms of the other variable and substitute it into the perimeter formula to get a single variable equation.
Let's solve for L in terms of W from the area equation:
A = L * W
75 = L * W (equation 1)
Solving for L, we get:
L = 75 / W (equation 2)
Now substitute this equation 2 into the perimeter formula:
P = 2L + 2W
P = 2(75 / W) + 2W
P = 150 / W + 2W
To minimize the perimeter, we need to find the value of W that will give us the lowest possible value for P. We can do this by taking the derivative of P with respect to W and setting it equal to 0.
dP/dW = -150 / W^2 + 2
Setting this derivative equal to 0:
-150 / W^2 + 2 = 0
Solving for W:
150 / W^2 = 2
150 = 2W^2
W^2 = 75
W = √75
W ≈ 8.66 ft
Now substitute this value of W into equation 1 to find L:
75 = L * 8.66
L ≈ 8.66 ft
Therefore, to minimize the perimeter of the rectangular room with an area of 75 ft, the architect should design the room with dimensions approximately 8.66 ft by 8.66 ft.
To minimize the perimeter of a rectangular room with a given area, we can use calculus. Let's assume the length of the room is denoted by "l" and the width is denoted by "w".
The perimeter of a rectangle is given by the formula:
P = 2l + 2w
The area of the rectangle is given by the formula:
A = l * w
We are given that the area, A, should be 75 square feet. So, we have the equation:
l * w = 75
To minimize the perimeter, we need to express the perimeter equation only in terms of one variable (either l or w) using the area equation. We can express l in terms of w by dividing both sides of the area equation by w:
l = 75 / w
Now substitute this expression for l into the perimeter equation:
P = 2(75 / w) + 2w
To find the minimum value of P, we need to take the derivative of P with respect to w, set it equal to zero, and solve for w. Let's differentiate P with respect to w:
dP/dw = -150 / w^2 + 2
Setting the derivative equal to zero, we have:
-150 / w^2 + 2 = 0
Now solve for w:
-150 / w^2 = -2
w^2 = 75
w = √75 ≈ 8.66 (rounded to two decimal places)
Now substitute this value of w back into the equation of the length:
l = 75 / 8.66 ≈ 8.66 (rounded to two decimal places)
Therefore, to minimize the perimeter of the rectangular room with an area of 75 square feet, the architect should use dimensions of approximately 8.66 ft by 8.66 ft.