To express the total number of gallons of water in the pool as a function of time \( t \) (in hours), we can use the formula you provided.
Given:
- Initial number of gallons in the pool: 250 gallons
- Rate of change (filling rate): 500 gallons per hour
- Time in hours: \( t \)
Formula:
The formula to find the number of gallons \( g \) in the pool after \( t \) hours is:
\[ g = \text{Initial gallons} + \text{(Rate of change)} \times t \]
Substituting the values:
- Initial gallons = 250
- Rate of change = 500
- Time = \( t \)
Therefore, the equation becomes:
\[ g = 250 + 500t \]
Values for \( t \) from 0 to 10:
Now, let's calculate the value of \( g \) for each \( t \) from 0 to 10:
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When \( t = 0 \): \[ g = 250 + 500 \times 0 = 250 \]
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When \( t = 1 \): \[ g = 250 + 500 \times 1 = 750 \]
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When \( t = 2 \): \[ g = 250 + 500 \times 2 = 1,250 \]
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When \( t = 3 \): \[ g = 250 + 500 \times 3 = 1,750 \]
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When \( t = 4 \): \[ g = 250 + 500 \times 4 = 2,250 \]
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When \( t = 5 \): \[ g = 250 + 500 \times 5 = 2,750 \]
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When \( t = 6 \): \[ g = 250 + 500 \times 6 = 3,250 \]
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When \( t = 7 \): \[ g = 250 + 500 \times 7 = 3,750 \]
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When \( t = 8 \): \[ g = 250 + 500 \times 8 = 4,250 \]
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When \( t = 9 \): \[ g = 250 + 500 \times 9 = 4,750 \]
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When \( t = 10 \): \[ g = 250 + 500 \times 10 = 5,250 \]
Summary:
Here is a summary of the values in a table format:
| Time (hours) | Gallons of Water | |--------------|------------------| | 0 | 250 | | 1 | 750 | | 2 | 1,250 | | 3 | 1,750 | | 4 | 2,250 | | 5 | 2,750 | | 6 | 3,250 | | 7 | 3,750 | | 8 | 4,250 | | 9 | 4,750 | | 10 | 5,250 |
Thus, the gallons of water in the pool can be calculated using the equation \( g = 250 + 500t \) for values of \( t \) from 0 to 10 hours.
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