Question

the sunrise donut shop sells boxes of one dozen glazed donuts. on average, it sells 125 boxes per day. the company is considering two new designs for the boxes. in box a, the donuts lay flat, while in box b, the donuts are stacked on their sides. both designs use the same material, which costs $0.35 per square foot.

box a
box b

which design would cost the shop less?
box a box b

to the nearest dollar, how much less, on average, would it cost the shop per day to use that design?

Explanation:

Step1: Calculate Box A surface area

First, identify the net of Box A: it has 6 rectangles.

• 2 rectangles: $14 \text{ in} \times 10.5 \text{ in}$
• 2 rectangles: $14 \text{ in} \times 1.5 \text{ in}$
• 2 rectangles: $10.5 \text{ in} \times 1.5 \text{ in}$

Surface Area of Box A:

$$\begin{align*} SA_A &= 2(14 \times 10.5) + 2(14 \times 1.5) + 2(10.5 \times 1.5) \\ &= 2(147) + 2(21) + 2(15.75) \\ &= 294 + 42 + 31.5 \\ &= 367.5 \text{ square inches} \end{align*}$$

Convert to square feet (1 square foot = 144 square inches):
$$SA_A = \frac{367.5}{144} \approx 2.5521 \text{ square feet}$$

Step2: Calculate Box B surface area

Identify the net of Box B: it has 6 rectangles.

• 2 rectangles: $9 \text{ in} \times 7 \text{ in}$
• 2 rectangles: $9 \text{ in} \times 3.5 \text{ in}$
• 2 rectangles: $7 \text{ in} \times 3.5 \text{ in}$

Surface Area of Box B:

$$\begin{align*} SA_B &= 2(9 \times 7) + 2(9 \times 3.5) + 2(7 \times 3.5) \\ &= 2(63) + 2(31.5) + 2(24.5) \\ &= 126 + 63 + 49 \\ &= 238 \text{ square inches} \end{align*}$$

Convert to square feet:
$$SA_B = \frac{238}{144} \approx 1.6528 \text{ square feet}$$

Step3: Calculate cost per box for each

Cost per box = Surface Area $\times \$0.35$ per square foot

• Cost of Box A: $2.5521 \times 0.35 \approx \$0.8932$
• Cost of Box B: $1.6528 \times 0.35 \approx \$0.5785$

Step4: Calculate daily total cost for each

Daily sales = 125 boxes

• Daily cost for Box A: $125 \times 0.8932 = \$111.65$
• Daily cost for Box B: $125 \times 0.5785 = \$72.31$

Step5: Calculate daily cost difference

$$111.65 - 72.31 = 39.34 \approx \$39$$

Answer:

1. Box B
2. $\$39$