Question
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 includes the siding. How many windows must a house have to make the total cost from Company A be less expensive than Company B?
Answers
GPT-4o mini
To determine how many windows a house must have for Company A to be cheaper than Company B, we can set up the following cost equations for both companies:
- For Company A:
\[
\text{Cost}_A = 312 + 12w
\]
where \( w \) is the number of windows.
- For Company B:
\[
\text{Cost}_B = 36w
\]
We want to find the number of windows \( w \) such that:
\[
\text{Cost}_A < \text{Cost}_B
\]
This can be written as:
\[
312 + 12w < 36w
\]
Now, we can solve this inequality step-by-step.
1. Subtract \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
This simplifies to:
\[
312 < 24w
\]
2. Divide both sides by 24:
\[
\frac{312}{24} < w
\]
Simplifying the left side:
\[
13 < w
\]
or
\[
w > 13
\]
This means that the house must have more than 13 windows for the total cost from Company A to be less expensive than that from Company B.
Thus, the minimum number of windows for Company A to be cheaper is \( \mathbf{14} \).
- For Company A:
\[
\text{Cost}_A = 312 + 12w
\]
where \( w \) is the number of windows.
- For Company B:
\[
\text{Cost}_B = 36w
\]
We want to find the number of windows \( w \) such that:
\[
\text{Cost}_A < \text{Cost}_B
\]
This can be written as:
\[
312 + 12w < 36w
\]
Now, we can solve this inequality step-by-step.
1. Subtract \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
This simplifies to:
\[
312 < 24w
\]
2. Divide both sides by 24:
\[
\frac{312}{24} < w
\]
Simplifying the left side:
\[
13 < w
\]
or
\[
w > 13
\]
This means that the house must have more than 13 windows for the total cost from Company A to be less expensive than that from Company B.
Thus, the minimum number of windows for Company A to be cheaper is \( \mathbf{14} \).
answers are:
312 - 12w > 36w
312 + 12w < 36w
12w + 36w > 312
36w - 12w < 312
312 - 12w > 36w
312 + 12w < 36w
12w + 36w > 312
36w - 12w < 312
GPT-4o mini
Let's analyze the information you've provided regarding the inequalities in order to evaluate the comparison of costs between Company A and Company B.
Since we need to find how many windows \( w \) must be installed for Company A to be cheaper than Company B, we can start from the correct setup.
The costs we have previously established are:
- For **Company A**:
\[
\text{Cost}_A = 312 + 12w
\]
- For **Company B**:
\[
\text{Cost}_B = 36w
\]
We need to establish the inequality:
\[
312 + 12w < 36w
\]
Now let's manipulate this inequality step by step:
1. **Isolate w** by subtracting \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
This simplifies to:
\[
312 < 24w
\]
2. **Divide both sides by 24** to solve for \( w \):
\[
\frac{312}{24} < w
\]
Simplifying the left side gives:
\[
13 < w
\]
or equivalently:
\[
w > 13
\]
This indicates that for Company A's total cost to be less than Company B's total cost, the number of windows \( w \) must be greater than 13. Therefore, the smallest integer that satisfies this is:
\[
w = 14
\]
Thus the correct answer remains: **A house must have at least 14 windows** for Company A to be less expensive than Company B.
If you have additional conditions or variations in the inequalities you want to explore, please let me know!
Since we need to find how many windows \( w \) must be installed for Company A to be cheaper than Company B, we can start from the correct setup.
The costs we have previously established are:
- For **Company A**:
\[
\text{Cost}_A = 312 + 12w
\]
- For **Company B**:
\[
\text{Cost}_B = 36w
\]
We need to establish the inequality:
\[
312 + 12w < 36w
\]
Now let's manipulate this inequality step by step:
1. **Isolate w** by subtracting \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
This simplifies to:
\[
312 < 24w
\]
2. **Divide both sides by 24** to solve for \( w \):
\[
\frac{312}{24} < w
\]
Simplifying the left side gives:
\[
13 < w
\]
or equivalently:
\[
w > 13
\]
This indicates that for Company A's total cost to be less than Company B's total cost, the number of windows \( w \) must be greater than 13. Therefore, the smallest integer that satisfies this is:
\[
w = 14
\]
Thus the correct answer remains: **A house must have at least 14 windows** for Company A to be less expensive than Company B.
If you have additional conditions or variations in the inequalities you want to explore, please let me know!
pick an answer.
GPT-4o mini
Based on the analysis, the answer is:
**14 windows**
A house must have at least 14 windows for the cost from Company A to be less expensive than the cost from Company B.
**14 windows**
A house must have at least 14 windows for the cost from Company A to be less expensive than the cost from Company B.