An architect is designing the floors for a hotel. Each floor is to be rectangular and is allotted 720 ft of security piping around walls outside the rooms. What dimensions of the atrium will allow an atrium at the bottom to have maximum area?

Probably a square.

720/4 = 180 ft. on each side

To find the dimensions of the atrium that will allow the maximum area, we need to understand the constraints and consider the available information.

Let's denote the length and width of the atrium as L and W, respectively. The perimeter of the atrium is then given as:

P = 2(L + W)

The problem states that each floor is allotted 720 ft of security piping around the walls outside the rooms. Since the atrium is located at the bottom, there are no walls outside the rooms on the lower floor. Hence, all the security piping will be used to enclose the perimeter of the atrium.

Therefore, we have:

P = 720 ft

Substituting the value of P in terms of L and W:

2(L + W) = 720

Now, we need to express the area of the atrium in terms of L and W. The area of a rectangle is given by:

A = L * W

To find the dimensions that maximize the area, we need to find the critical points of the area function. We can do this by taking the derivative of A with respect to L and setting it to zero. Let's differentiate:

dA/dL = W

Setting dA/dL = 0:

W = 0

Since W cannot be zero (as it represents the width of the atrium), we can conclude that there are no critical points within the given constraints.

Hence, to maximize the area of the atrium, we need to consider the extreme values of L and W. In this case, the extreme values occur when L + W is minimized while still satisfying the perimeter constraint. Since the perimeter is fixed at 720 ft, we can rewrite the perimeter equation as:

L + W = 360

Now, we have a function with only one variable (L or W). Let's solve for L in terms of W:

L = 360 - W

Substituting this value of L in the area equation:

A = (360 - W) * W = 360W - W^2

This is a quadratic equation with a negative leading coefficient (-1), which means the graph of A will be a downward-opening parabola. The vertex of the parabola represents the maximum point of the area.

To find the vertex, we can use the formula:

W_vertex = -b / (2a)

In this case, a = -1 and b = 360. Substituting these values, we get:

W_vertex = -360 / (2 * -1) = 180

Therefore, the width of the atrium that maximizes the area is 180 ft.

We can substitute this value back into the perimeter equation to find the corresponding length:

L + 180 = 360
L = 180 ft

Hence, the dimensions of the atrium that allow for maximum area are 180 ft by 180 ft.