Question
Choose the inequality that matches this situation.
Company A charges $312 to power-wash the siding of a house plus $12 for each window. Company B charges $36 per window, and the price includs the siding. How many windows must a house have to make the total cost from Company A be less expensive than Company B?
explain also
answers for this is: 312-12w>36w
312+12w<36w
12w+36w>312
36w-12w<312
Company A charges $312 to power-wash the siding of a house plus $12 for each window. Company B charges $36 per window, and the price includs the siding. How many windows must a house have to make the total cost from Company A be less expensive than Company B?
explain also
answers for this is: 312-12w>36w
312+12w<36w
12w+36w>312
36w-12w<312
Answers
GPT-4o mini
To determine which inequality matches the situation, let's first express the total costs for both companies in terms of the number of windows, denoted as \( w \).
**Company A:**
- The cost is a fixed charge of $312 for power-washing the siding plus $12 for each window. Therefore, the total cost for Company A can be expressed as:
\[ \text{Cost}_A = 312 + 12w \]
**Company B:**
- Company B charges $36 per window, and this includes the siding. Thus, the total cost for Company B is:
\[ \text{Cost}_B = 36w \]
We want to find the number of windows \( w \) such that the total cost from Company A is less expensive than Company B. This can be expressed mathematically as:
\[ \text{Cost}_A < \text{Cost}_B \]
Substituting the cost expressions:
\[ 312 + 12w < 36w \]
To simplify this inequality:
1. Subtract \( 12w \) from both sides:
\[ 312 < 36w - 12w \]
\[ 312 < 24w \]
2. Divide both sides by 24:
\[ 13 < w \]
or equivalently,
\[ w > 13 \]
This tells us that for the cost from Company A to be less than that from Company B, the house must have more than 13 windows.
Now, looking at your provided inequalities:
- **312 - 12w > 36w** (not correct; rearranging would lead to a wrong relationship)
- **312 + 12w < 36w** (this is correct; it matches what we derived)
- **12w + 36w > 312** (not correct; simplifies to 48w > 312, but this is not what we need)
- **36w - 12w < 312** (not directly useful since it does not compare the costs correctly)
The correct inequality corresponding to the situation is:
**312 + 12w < 36w**
This shows that the cost from Company A must be less than that from Company B for the number of windows \( w \).
**Company A:**
- The cost is a fixed charge of $312 for power-washing the siding plus $12 for each window. Therefore, the total cost for Company A can be expressed as:
\[ \text{Cost}_A = 312 + 12w \]
**Company B:**
- Company B charges $36 per window, and this includes the siding. Thus, the total cost for Company B is:
\[ \text{Cost}_B = 36w \]
We want to find the number of windows \( w \) such that the total cost from Company A is less expensive than Company B. This can be expressed mathematically as:
\[ \text{Cost}_A < \text{Cost}_B \]
Substituting the cost expressions:
\[ 312 + 12w < 36w \]
To simplify this inequality:
1. Subtract \( 12w \) from both sides:
\[ 312 < 36w - 12w \]
\[ 312 < 24w \]
2. Divide both sides by 24:
\[ 13 < w \]
or equivalently,
\[ w > 13 \]
This tells us that for the cost from Company A to be less than that from Company B, the house must have more than 13 windows.
Now, looking at your provided inequalities:
- **312 - 12w > 36w** (not correct; rearranging would lead to a wrong relationship)
- **312 + 12w < 36w** (this is correct; it matches what we derived)
- **12w + 36w > 312** (not correct; simplifies to 48w > 312, but this is not what we need)
- **36w - 12w < 312** (not directly useful since it does not compare the costs correctly)
The correct inequality corresponding to the situation is:
**312 + 12w < 36w**
This shows that the cost from Company A must be less than that from Company B for the number of windows \( w \).