An architect is designing an atrium for a hotel. The Atrium is to be rectangular with a perimeter of 728ft of brass piping. What dimensions will maximize the area of the atrium?
p = 2 L + 2 w = 728 so L + w = 364 and L = (364-w)
A = L w
A = (364-w)w = 364 w - w^2
dA/dw = 0 at max or min = 364 -2w
so
w = 182
L = 364-w = 182
LOL, a square
or if you do not know calculus
w^2 - 364 w = -A
complete square to find vertex
w^2 - 364 w + (182)^2 = -A+182^2
(w-182)^2 = -A+182^2
vertex at w = 182 again
To maximize the area of the atrium, we need to find the dimensions that will maximize the rectangle's area. Let's denote the length of the atrium as L and the width as W.
Based on the information given, we know that the perimeter of the atrium is 728ft. The perimeter of a rectangle is given by the formula:
Perimeter = 2L + 2W
Since all sides of the atrium are made of brass piping, we can write the equation as:
728 = 2L + 2W
Simplifying this equation, we get:
364 = L + W
To maximize the area, we need to find the dimensions that satisfy this equation while maximizing the product of L and W, which represents the area.
To solve this problem, we can use the method of differentiation. Taking the derivative of the area equation, we get:
d(area) / dL = 0
d(area) / dW = 0
Let's take the derivative of the area equation with respect to L:
d(area) / dL = W
And now, let's take the derivative of the area equation with respect to W:
d(area) / dW = L
Since both of these derivatives need to be equal to zero to find the maximum area, we can set them equal to zero:
W = 0
L = 0
This implies that L and W must equal zero, which is not possible for the dimensions of the atrium.
Therefore, there is not enough information given to determine the dimensions of the rectangle that will maximize the area.
To find the dimensions that will maximize the area of the atrium, we need to use calculus optimization techniques. Let's solve the problem step by step:
Step 1: Understanding the problem
The problem states that the perimeter of the atrium is given by 728 ft of brass piping. Since the atrium is rectangular, we need to find the dimensions (length and width) that maximize its area.
Step 2: Set up the problem
Let's define the length of the atrium as "l" and the width as "w." The perimeter of a rectangle is given by the formula P = 2l + 2w. Since we know that the perimeter is 728 ft, we can write the equation as:
2l + 2w = 728
Step 3: Simplify the equation
Since we are trying to maximize the area, which is given by A = l * w, we can solve the equation from Step 2 for one variable and substitute it into the area equation. Let's solve the perimeter equation for w:
2l + 2w = 728
2w = 728 - 2l
w = 364 - l/2
Step 4: Express the area in terms of one variable
Substitute the expression for w obtained in Step 3 into the area equation A = l * w:
A = l * (364 - l/2)
A = l * (364 - 0.5l)
A = 364l - 0.5l^2
Step 5: Maximize the area
Now, to find the maximum area, we need to find the critical points of the area function. Take the derivative of the area function with respect to l and set it equal to zero:
dA/dl = 364 - l = 0
Solve for l:
l = 364
Step 6: Determine the dimensions
Substitute the value of l obtained in Step 5 back into the equation from Step 3 to find the value of w:
w = 364 - l/2
w = 364 - 364/2
w = 364 - 182
w = 182
Therefore, the dimensions that will maximize the area of the atrium are: length = 364 ft and width = 182 ft.