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 the correct total charges for each set of cube-shaped boxes, let's calculate the charges based on the provided formulas.
The surface area \( S \) of a cube with side length \( s \) is given by:
\[ S = 6s^2 \]
For a side length \( x \) of at least 5 inches, the charge is calculated as:
\[ \text{Charge} = 0.05 \cdot S = 0.05 \cdot 6x^2 = 0.30x^2 \]
For a side length \( y \) of less than 5 inches, the charge is calculated as:
\[ \text{Charge} = 0.04 \cdot S = 0.04 \cdot 6y^2 = 0.24y^2 \]
Now, we will evaluate each option:
**A. Noella wraps two cube-shaped boxes of side lengths 6 inches and 8 inches for a total of $30.00.**
1. For 6 inches:
\[
\text{Charge} = 0.30 \cdot 6^2 = 0.30 \cdot 36 = 10.80
\]
2. For 8 inches:
\[
\text{Charge} = 0.30 \cdot 8^2 = 0.30 \cdot 64 = 19.20
\]
Total charge:
\[
10.80 + 19.20 = 30.00
\]
This option is correct.
---
**B. Noella wraps two cube-shaped boxes of side lengths 7 inches and 5.5 inches for a total of $21.96.**
1. For 7 inches:
\[
\text{Charge} = 0.30 \cdot 7^2 = 0.30 \cdot 49 = 14.70
\]
2. For 5.5 inches:
\[
\text{Charge} = 0.30 \cdot 5.5^2 = 0.30 \cdot 30.25 = 9.075
\]
Total charge:
\[
14.70 + 9.075 = 23.775
\]
This option is incorrect.
---
**C. Noella wraps two cube-shaped boxes of side lengths 4.5 inches and 5 inches for a total of $12.36.**
1. For 4.5 inches:
\[
\text{Charge} = 0.24 \cdot (4.5^2) = 0.24 \cdot 20.25 = 4.86
\]
2. For 5 inches:
\[
\text{Charge} = 0.30 \cdot (5^2) = 0.30 \cdot 25 = 7.50
\]
Total charge:
\[
4.86 + 7.50 = 12.36
\]
This option is correct.
---
**D. Noella wraps two cube-shaped boxes of side lengths 5 inches and 4 inches for a total of $11.34.**
1. For 5 inches:
\[
\text{Charge} = 0.30 \cdot (5^2) = 0.30 \cdot 25 = 7.50
\]
2. For 4 inches:
\[
\text{Charge} = 0.24 \cdot (4^2) = 0.24 \cdot 16 = 3.84
\]
Total charge:
\[
7.50 + 3.84 = 11.34
\]
This option is correct.
---
**E. Noella wraps two cube-shaped boxes of side lengths 3.5 inches and 6 inches for a total of $12.32.**
1. For 3.5 inches:
\[
\text{Charge} = 0.24 \cdot (3.5^2) = 0.24 \cdot 12.25 = 2.94
\]
2. For 6 inches:
\[
\text{Charge} = 0.30 \cdot (6^2) = 0.30 \cdot 36 = 10.80
\]
Total charge:
\[
2.94 + 10.80 = 13.74
\]
This option is incorrect.
---
**Summary of correct options:**
- A: Correct! (Total: $30.00)
- C: Correct! (Total: $12.36)
- D: Correct! (Total: $11.34)
Thus, the correct options to represent the amounts charged are **A, C, and D.**
The surface area \( S \) of a cube with side length \( s \) is given by:
\[ S = 6s^2 \]
For a side length \( x \) of at least 5 inches, the charge is calculated as:
\[ \text{Charge} = 0.05 \cdot S = 0.05 \cdot 6x^2 = 0.30x^2 \]
For a side length \( y \) of less than 5 inches, the charge is calculated as:
\[ \text{Charge} = 0.04 \cdot S = 0.04 \cdot 6y^2 = 0.24y^2 \]
Now, we will evaluate each option:
**A. Noella wraps two cube-shaped boxes of side lengths 6 inches and 8 inches for a total of $30.00.**
1. For 6 inches:
\[
\text{Charge} = 0.30 \cdot 6^2 = 0.30 \cdot 36 = 10.80
\]
2. For 8 inches:
\[
\text{Charge} = 0.30 \cdot 8^2 = 0.30 \cdot 64 = 19.20
\]
Total charge:
\[
10.80 + 19.20 = 30.00
\]
This option is correct.
---
**B. Noella wraps two cube-shaped boxes of side lengths 7 inches and 5.5 inches for a total of $21.96.**
1. For 7 inches:
\[
\text{Charge} = 0.30 \cdot 7^2 = 0.30 \cdot 49 = 14.70
\]
2. For 5.5 inches:
\[
\text{Charge} = 0.30 \cdot 5.5^2 = 0.30 \cdot 30.25 = 9.075
\]
Total charge:
\[
14.70 + 9.075 = 23.775
\]
This option is incorrect.
---
**C. Noella wraps two cube-shaped boxes of side lengths 4.5 inches and 5 inches for a total of $12.36.**
1. For 4.5 inches:
\[
\text{Charge} = 0.24 \cdot (4.5^2) = 0.24 \cdot 20.25 = 4.86
\]
2. For 5 inches:
\[
\text{Charge} = 0.30 \cdot (5^2) = 0.30 \cdot 25 = 7.50
\]
Total charge:
\[
4.86 + 7.50 = 12.36
\]
This option is correct.
---
**D. Noella wraps two cube-shaped boxes of side lengths 5 inches and 4 inches for a total of $11.34.**
1. For 5 inches:
\[
\text{Charge} = 0.30 \cdot (5^2) = 0.30 \cdot 25 = 7.50
\]
2. For 4 inches:
\[
\text{Charge} = 0.24 \cdot (4^2) = 0.24 \cdot 16 = 3.84
\]
Total charge:
\[
7.50 + 3.84 = 11.34
\]
This option is correct.
---
**E. Noella wraps two cube-shaped boxes of side lengths 3.5 inches and 6 inches for a total of $12.32.**
1. For 3.5 inches:
\[
\text{Charge} = 0.24 \cdot (3.5^2) = 0.24 \cdot 12.25 = 2.94
\]
2. For 6 inches:
\[
\text{Charge} = 0.30 \cdot (6^2) = 0.30 \cdot 36 = 10.80
\]
Total charge:
\[
2.94 + 10.80 = 13.74
\]
This option is incorrect.
---
**Summary of correct options:**
- A: Correct! (Total: $30.00)
- C: Correct! (Total: $12.36)
- D: Correct! (Total: $11.34)
Thus, the correct options to represent the amounts charged are **A, C, and D.**