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To find Conor's total cost for a ticket, we need to consider both the ticket price and the amusement tax. The amusement tax is 15% of the ticket price \( t \). Thus, we can express the total cost in two ways:
- Total cost without tax: \( C = t \)
- Total cost including the tax: \( C = t + 0.15t \)
We can simplify the second expression:
\[ C = t + 0.15t = 1.15t \]
Therefore, the two expressions for Conor's total cost in terms of \( t \) are:
- \( C = t \)
- \( C = 1.15t \)
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Let's choose a value for \( t \) between $12 and $13. For instance, we'll choose \( t = 12.50 \).
Now we can use the expression for the total cost including tax:
\[ C = 1.15t \]
Substituting \( t = 12.50 \):
\[ C = 1.15 \times 12.50 \]
Now, let's calculate:
\[ C = 1.15 \times 12.50 = 14.375 \]
Rounding to the nearest penny:
\[ C \approx 14.38 \]
Thus, Conor's total cost is $14.38.
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We are told that Conor's total cost \( C \) is greater than $17.50 and less than $20. So, we can set up the inequality using the total cost including tax:
\[ 17.50 < 1.15t < 20 \]
To solve for \( t \), we can divide the entire inequality by 1.15:
\[ \frac{17.50}{1.15} < t < \frac{20}{1.15} \]
Now, we calculate the bounds:
\[ \frac{17.50}{1.15} \approx 15.2174 \] \[ \frac{20}{1.15} \approx 17.3913 \]
Therefore, the inequality simplifies to:
\[ 15.2174 < t < 17.3913 \]
With the condition that \( t \) must be greater than $12 and less than $13, we can find values greater than 15.2174 and less than 17.3913.
A valid value for \( t \) that meets this criterion could be \( t = 16.00 \).
Thus, a possible value for \( t \) is $16.00.
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