- Analyze the given function \(E(q) = 5 + 4q\).
- Compute the function outputs for inputs \(q=0\), \(q=1\), and \(q=2\).
- The values are \(E(0)=5\), \(E(1)=9\), and \(E(2)=13\).
- Compare these results with the provided tables to find the correct one, which is the second table: \(\boxed{\text{Table 2}}\).
Explanation
- Analysis of the Expense Function
We are given the expense function \(E(q) = 5 + 4q\), where \(q\) is the quantity and \(E(q)\) represents the total expenses. To identify the correct table, we need to calculate the values of \(E(q)\) for the input quantities \(q = 0\), \(q = 1\), and \(q = 2\) and compare these results with the values provided in the given tables.
- Calculating Expected Values
We substitute the given values of \(q\) into the function \(E(q) = 5 + 4q\):
- For \(q = 0\): \(E(0) = 5 + 4(0) = 5 + 0 = 5\)
- For \(q = 1\): \(E(1) = 5 + 4(1) = 5 + 4 = 9\)
- For \(q = 2\): \(E(2) = 5 + 4(2) = 5 + 8 = 13\)
Thus, the expected sequence of values for \(E(q)\) is \(5, 9, 13\) for the quantities \(0, 1, 2\) respectively.
- Comparing Results and Identifying the Correct Table
Now, we compare our calculated values \((E(0)=5, E(1)=9, E(2)=13)\) with the tables:
- Table 1: \(q=0 \to 4\), \(q=1 \to 8\), \(q=2 \to 12\) (Incorrect)
- Table 2: \(q=0 \to 5\), \(q=1 \to 9\), \(q=2 \to 13\) (Matches our calculations)
- Table 3: \(q=0 \to 5\), \(q=1 \to 7\), \(q=2 \to 9\) (Incorrect)
- Table 4: \(q=0 \to 5\), \(q=1 \to 8\), \(q=2 \to 11\) (Incorrect)
Comparing these, only the second table correctly corresponds to the function \(E(q) = 5 + 4q\).
Examples
The function \(E(q) = 5 + 4q\) is a great example of a fixed cost plus a variable cost model. If you are starting a small lemonade stand, your fixed cost (the initial investment for signs and a pitcher) is \(\$5\), and each cup costs you \(\$4\) in ingredients. This simple linear equation helps you quickly calculate your total expenses as you decide how many cups to produce, ensuring your pricing covers your costs.