Elaine has a customer who needs a box with a volume of
12 cubic inches. Th e customer wants to know what size box is
the least expensive to buy.
• Th e price will be based on surface area. Th e supplier
charges _ 21_ cent per square inch.
• Th e dimensions of the box are whole numbers.
What size box do you recommend that Elaine’s customer buy?
Write a report to Elaine that shows your work and explains
your recommendation.
To recommend the most cost-effective box size for Elaine's customer, we need to mathematically determine the dimensions that provide the smallest surface area. The price is based on the surface area, and the supplier charges 21 cents per square inch. The dimensions of the box are whole numbers.
To begin, let's assume the dimensions of the box are length (L), width (W), and height (H) in inches. We need to find a combination of L, W, and H that satisfies the volume requirement of 12 cubic inches while minimizing the surface area.
The formula for the volume of a rectangular box is V = L * W * H, and in this case, V = 12 cubic inches. We know that L, W, and H are whole numbers.
Next, we need to find the formula for the surface area of the box. A rectangular box has six faces, and the surface area (SA) is the sum of the areas of these faces. For simplicity, let's denote the top and bottom faces as A1, the front and back faces as A2, and the left and right faces as A3.
The surface area formula is SA = 2(A1 + A2 + A3). To find the areas A1, A2, and A3, we relate them to the dimensions L, W, and H.
A1 = L * W
A2 = L * H
A3 = W * H
Now, let's express the surface area formula in terms of L, W, and H:
SA = 2(L * W + L * H + W * H)
To minimize the surface area, we'll use a method called "partial differentiation." However, since the dimensions must be whole numbers, we will use a systematic approach instead.
We can assume that L ≤ W ≤ H to avoid redundancy, as switching the values will give the same dimensions with a different order.
Now, we will find the smallest possible surface area by evaluating all possible combinations.
First, we consider the case where L = 1. Since W ≤ H, we start with W = 1:
SA = 2(1 * 1 + 1 * H + 1 * H)
SA = 2(2 + 2H)
Applying the volume requirement (V = 12), we can solve for H:
1 * 1 * H = 12
H = 12
Using these values (L = 1, W = 1, H = 12), we can calculate the surface area:
SA = 2(2 + 2 * 12) = 2(26) = 52 square inches.
Next, let's try L = 1 and W = 2. Using the volume requirement, we can find H:
1 * 2 * H = 12
H = 6
Using these values (L = 1, W = 2, H = 6):
SA = 2(2 + 1 * 6 + 2 * 6) = 2(2 + 6 + 12) = 2(20) = 40 square inches.
Continuing this process, we find the following dimensions and respective surface areas:
L = 1, W = 3, H = 4, SA = 38 square inches
L = 2, W = 2, H = 3, SA = 32 square inches
L = 3, W = 3, H = 2, SA = 42 square inches
L = 4, W = 2, H = 2, SA = 36 square inches
L = 6, W = 1, H = 2, SA = 40 square inches
L = 12, W = 1, H = 1, SA = 26 square inches
From these calculations, we can conclude that the dimensions L = 12, W = 1, and H = 1 result in the smallest surface area of 26 square inches. Therefore, this is the least expensive box to buy for Elaine's customer.
To support our recommendation, we can calculate the cost based on the surface area and supplier's pricing:
Cost = Surface Area * Price per square inch
Cost = 26 * $0.21
Cost = $5.46
So, Elaine's customer should buy a box with dimensions 12 x 1 x 1 inches, which has the least surface area and will cost $5.46.