Question
Noella wraps cube-shaped boxes at her gift shop. A customer who wants to wrap a cube-shaped box with a side length x of at least 5 inches is charged 5 cents times the surface area of the box, which totals 0.05 • 6 • x2 dollars. On the other hand, a customer who wants to wrap a cube-shaped box with a side length y of less than 5 inches is charged 4 cents times the surface area of the box, which totals 0.04 • 6 • y2 dollars.
Which of the following correctly represents the amount that Noella charges to wrap each set of cube-shaped boxes? Select all that apply.
A.
Noella wraps two cube-shaped boxes of side lengths 6 inches and 8 inches for a total of $30.00.
B.
Noella wraps two cube-shaped boxes of side lengths 7 inches and 5.5 inches for a total of $21.96.
C.
Noella wraps two cube-shaped boxes of side lengths 4.5 inches and 5 inches for a total of $12.36
D.
Noella wraps two cube-shaped boxes of side lengths 5 inches and 4 inches for a total of $11.34.
E.
Noella wraps two cube-shaped boxes of side lengths 3.5 inches and 6 inches for a total of $12.32.
Which of the following correctly represents the amount that Noella charges to wrap each set of cube-shaped boxes? Select all that apply.
A.
Noella wraps two cube-shaped boxes of side lengths 6 inches and 8 inches for a total of $30.00.
B.
Noella wraps two cube-shaped boxes of side lengths 7 inches and 5.5 inches for a total of $21.96.
C.
Noella wraps two cube-shaped boxes of side lengths 4.5 inches and 5 inches for a total of $12.36
D.
Noella wraps two cube-shaped boxes of side lengths 5 inches and 4 inches for a total of $11.34.
E.
Noella wraps two cube-shaped boxes of side lengths 3.5 inches and 6 inches for a total of $12.32.
Answers
GPT-4o mini
To determine which options are correct regarding Noella's charges for wrapping cube-shaped boxes, we first need to calculate the costs based on the formulas given for the two cases:
1. **For side length \( x \) (where \( x \geq 5 \)):**
\[
\text{Charge} = 0.05 \times 6 \times x^2 = 0.3 \times x^2
\]
2. **For side length \( y \) (where \( y < 5 \)):**
\[
\text{Charge} = 0.04 \times 6 \times y^2 = 0.24 \times y^2
\]
Now, we will compute the total charges based on the specified side lengths for each option.
**A. Side lengths 6 inches and 8 inches**
- For 6 inches:
\[
\text{Charge} = 0.3 \times 6^2 = 0.3 \times 36 = 10.80
\]
- For 8 inches:
\[
\text{Charge} = 0.3 \times 8^2 = 0.3 \times 64 = 19.20
\]
- Total charge:
\[
10.80 + 19.20 = 30.00
\]
This option is correct.
**B. Side lengths 7 inches and 5.5 inches**
- For 7 inches:
\[
\text{Charge} = 0.3 \times 7^2 = 0.3 \times 49 = 14.70
\]
- For 5.5 inches:
\[
\text{Charge} = 0.3 \times 5.5^2 = 0.3 \times 30.25 = 9.075
\]
- Total charge:
\[
14.70 + 9.075 = 23.775
\]
Since $23.775 does not equal $21.96, this option is incorrect.
**C. Side lengths 4.5 inches and 5 inches**
- For 4.5 inches:
\[
\text{Charge} = 0.24 \times 4.5^2 = 0.24 \times 20.25 = 4.86
\]
- For 5 inches:
\[
\text{Charge} = 0.3 \times 5^2 = 0.3 \times 25 = 7.50
\]
- Total charge:
\[
4.86 + 7.50 = 12.36
\]
This option is correct.
**D. Side lengths 5 inches and 4 inches**
- For 5 inches:
\[
\text{Charge} = 0.3 \times 5^2 = 0.3 \times 25 = 7.50
\]
- For 4 inches:
\[
\text{Charge} = 0.24 \times 4^2 = 0.24 \times 16 = 3.84
\]
- Total charge:
\[
7.50 + 3.84 = 11.34
\]
This option is correct.
**E. Side lengths 3.5 inches and 6 inches**
- For 3.5 inches:
\[
\text{Charge} = 0.24 \times 3.5^2 = 0.24 \times 12.25 = 2.94
\]
- For 6 inches:
\[
\text{Charge} = 0.3 \times 6^2 = 0.3 \times 36 = 10.80
\]
- Total charge:
\[
2.94 + 10.80 = 13.74
\]
Since $13.74 does not equal $12.32, this option is incorrect.
**Correct Answers: A, C, D**.
1. **For side length \( x \) (where \( x \geq 5 \)):**
\[
\text{Charge} = 0.05 \times 6 \times x^2 = 0.3 \times x^2
\]
2. **For side length \( y \) (where \( y < 5 \)):**
\[
\text{Charge} = 0.04 \times 6 \times y^2 = 0.24 \times y^2
\]
Now, we will compute the total charges based on the specified side lengths for each option.
**A. Side lengths 6 inches and 8 inches**
- For 6 inches:
\[
\text{Charge} = 0.3 \times 6^2 = 0.3 \times 36 = 10.80
\]
- For 8 inches:
\[
\text{Charge} = 0.3 \times 8^2 = 0.3 \times 64 = 19.20
\]
- Total charge:
\[
10.80 + 19.20 = 30.00
\]
This option is correct.
**B. Side lengths 7 inches and 5.5 inches**
- For 7 inches:
\[
\text{Charge} = 0.3 \times 7^2 = 0.3 \times 49 = 14.70
\]
- For 5.5 inches:
\[
\text{Charge} = 0.3 \times 5.5^2 = 0.3 \times 30.25 = 9.075
\]
- Total charge:
\[
14.70 + 9.075 = 23.775
\]
Since $23.775 does not equal $21.96, this option is incorrect.
**C. Side lengths 4.5 inches and 5 inches**
- For 4.5 inches:
\[
\text{Charge} = 0.24 \times 4.5^2 = 0.24 \times 20.25 = 4.86
\]
- For 5 inches:
\[
\text{Charge} = 0.3 \times 5^2 = 0.3 \times 25 = 7.50
\]
- Total charge:
\[
4.86 + 7.50 = 12.36
\]
This option is correct.
**D. Side lengths 5 inches and 4 inches**
- For 5 inches:
\[
\text{Charge} = 0.3 \times 5^2 = 0.3 \times 25 = 7.50
\]
- For 4 inches:
\[
\text{Charge} = 0.24 \times 4^2 = 0.24 \times 16 = 3.84
\]
- Total charge:
\[
7.50 + 3.84 = 11.34
\]
This option is correct.
**E. Side lengths 3.5 inches and 6 inches**
- For 3.5 inches:
\[
\text{Charge} = 0.24 \times 3.5^2 = 0.24 \times 12.25 = 2.94
\]
- For 6 inches:
\[
\text{Charge} = 0.3 \times 6^2 = 0.3 \times 36 = 10.80
\]
- Total charge:
\[
2.94 + 10.80 = 13.74
\]
Since $13.74 does not equal $12.32, this option is incorrect.
**Correct Answers: A, C, D**.