To determine how many miles Nathan can drive without exceeding his budget, we start by calculating the total cost of renting the car including the mileage fee.
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Daily rental charge is $29.50.
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Total budget: $75.
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Remaining budget for mileage: \[ \text{Remaining budget} = \text{Total budget} - \text{Daily rental charge} \] \[ \text{Remaining budget} = 75 - 29.50 = 45.50 \]
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Cost per mile is $0.55. Now, we need to determine how many miles Nathan can drive with the remaining budget of $45.50. To find the maximum number of miles (\(x\)), we can set up the equation: \[ 0.55x = 45.50 \]
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Solving for \(x\): \[ x = \frac{45.50}{0.55} \] \[ x = 82.7272... \]
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Since Nathan can't drive a fractional mile, we round up to the next whole number because any partial mile would still incur a full charge. Therefore, he can drive: \[ x = 83 \text{ miles.} \]
Thus, Nathan can drive a maximum of 83 miles without exceeding his budget.