Let's break this down step by step.
Part a)
We need to write two expressions for Conor's total cost, including the amusement tax on the ticket.
- The cost of the ticket is \( t \) dollars.
- The amusement tax is 15% of \( t \), which can be written as \( 0.15t \).
Therefore, Conor’s total cost, including the tax, can be expressed with:
- Expression 1: \( t + 0.15t = 1.15t \)
- Expression 2: \( t + 0.15t = t(1 + 0.15) = 1.15t \)
Both expressions yield the same total cost: \( 1.15t \).
Part b)
Now, we need to choose a value for \( t \) that is greater than $12 and less than $13. A suitable value can be \( t = 12.50 \).
Now, we will use the expression \( 1.15t \) to calculate Conor's total cost:
\[ 1.15t = 1.15 \times 12.50 \] Calculating this, we have:
\[ 1.15 \times 12.50 = 14.375 \]
Rounding to the nearest penny, Conor’s total cost is:
\[ \text{Total Cost} = 14.38 \]
Part c)
Next, we are to find a value of \( t \) such that Conor’s total cost is greater than $17.50 and less than $20. To find this value, we can set up the inequality using our total cost expression:
\[ 17.50 < 1.15t < 20 \]
We can solve this inequality step by step.
- Solving the left side:
\[ 17.50 < 1.15t \]
Divide both sides by 1.15:
\[ \frac{17.50}{1.15} < t \]
Calculating it:
\[ t > 15.217 \quad (\text{approximately}) \]
- Now solve the right side:
\[ 1.15t < 20 \]
Again, divide both sides by 1.15:
\[ t < \frac{20}{1.15} \]
Calculating it:
\[ t < 17.391 \quad (\text{approximately}) \]
Combining the results, we find:
\[ 15.217 < t < 17.391 \]
Thus, any value for \( t \) within that range (for example, \( t = 16.00 \) or \( t = 16.50 \)) would ensure Conor’s total cost is between $17.50 and $20.
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