To express the number of gallons of water in the pool over time, we can use the formula you've given:
\[ g = 600 + \text{Change in gallons} \]
The change in gallons of water due to the leak can be calculated as follows:
\[ \text{Change in gallons} = \text{Change in gallons per hour} \times \text{Number of hours} = -100t \]
So we can substitute that into the equation for \( g \):
\[ g = 600 + (-100t) \]
This simplifies to:
\[ g = 600 - 100t \]
Now, we can calculate \( g \) for different values of \( t \) (the number of hours):
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When \( t = 0 \): \[ g = 600 - 100(0) = 600 \]
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When \( t = 1 \): \[ g = 600 - 100(1) = 500 \]
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When \( t = 2 \): \[ g = 600 - 100(2) = 400 \]
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When \( t = 3 \): \[ g = 600 - 100(3) = 300 \]
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When \( t = 4 \): \[ g = 600 - 100(4) = 200 \]
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When \( t = 5 \): \[ g = 600 - 100(5) = 100 \]
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When \( t = 6 \): \[ g = 600 - 100(6) = 0 \]
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When \( t = 7 \): \[ g = 600 - 100(7) = -100 \quad (\text{pool is empty}) \]
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When \( t = 8 \): \[ g = 600 - 100(8) = -200 \]
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When \( t = 9 \): \[ g = 600 - 100(9) = -300 \]
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When \( t = 10 \): \[ g = 600 - 100(10) = -400 \]
Based on these calculations, we see how the number of gallons in the swimming pool decreases over time until it becomes empty (at \( t = 6 \) hours) and progressively goes negative, indicating the theoretical water level if the leak continues beyond the pool's capacity.
The table of values for \( t \) and \( g \) would look as follows:
| Time in hours (t) | Gallons of water (g) | |--------------------|-----------------------| | 0 | 600 | | 1 | 500 | | 2 | 400 | | 3 | 300 | | 4 | 200 | | 5 | 100 | | 6 | 0 | | 7 | -100 | | 8 | -200 | | 9 | -300 | | 10 | -400 |
This provides a complete representation of the rate at which the water in the pool decreases with time given the leak.