Question

a shoe store owner is designing the layout of the displays in his store. the floor area for the store is 1,600 square feet. the owner wants to have circular displays for the shoes, each with a radius of 6 feet, and a 48 - square - foot rectangular cashiers station in the corner. customers should be able to easily walk around the shoe displays, so there must be at least a 4 - foot walkway around each. the owner started drawing the layout shown.
the distance between grid lines is 4 feet.
image of three circular shoe displays on a grid
which statement about the possible designs is true, given the owner’s requirements?

• three circular displays can be placed in one row, exactly as shown on the given grid.
• two circular displays can be placed so they are each the same distance from the cashier’s station.
• the maximum number of circular displays that can be used is 4.
• the maximum number of circular displays that can be used is 5.

Explanation:

Step1: Calculate store dimensions

The store is a square (since area is 1600 sq ft).
$$\sqrt{1600} = 40 \text{ feet}$$
So the store is 40ft × 40ft.

Step2: Calculate total width per display

Each circle has radius 6ft, so diameter $2\times6=12$ft. Add 4ft walkway on both sides: $12 + 4 + 4 = 20$ft per display (total space needed horizontally/vertically per display).

Step3: Evaluate 3 displays in one row

3 displays would need $3\times20 = 60$ft, but store is only 40ft wide. So this is impossible.

Step4: Calculate space for cashier + displays

Cashier takes 48 sq ft; assume it's placed in a corner, e.g., 6ft×8ft, which fits in the 40ft edge. Now check maximum displays:
Each display needs a 20ft×20ft "buffer zone" (for the circle + walkway). In a 40ft×40ft space, we can fit 2 displays along each side (2×20=40ft), so 4 total. Trying to fit 5 would require overlapping buffer zones, violating the walkway rule.

Step5: Evaluate equal distance claim

There is no guarantee two displays can be placed equidistant from the cashier while maintaining the 4ft walkway and fitting in the store, and this is not a necessary true statement compared to the maximum count.

Answer:

The maximum number of circular displays that can be used is 4.