- The total expense function is given by \(E(q) = 20q + 100\).
- Substitute \(q = 5\) into the function to get \(E(5) = 200\).
- Substitute \(q = 10\) into the function to get \(E(10) = 300\).
- Substitute \(q = 25\) into the function to get \(E(25) = 600\).
- Final values are: 200, 300, and 600.
Explanation
- Problem Analysis
The problem asks for the total expense incurred for different quantities of an item, given a specific cost function. The expense function is defined as \(E(q) = 20q + 100\), where \(q\) represents the quantity of the item produced, \(20\) represents the variable cost per item, and \(100\) represents the fixed costs. Our objective is to calculate the total expense for three specific quantities: \(q = 5\), \(q = 10\), and \(q = 25\).
- Calculation of Expenses
To find the total expense for each quantity, we substitute the value of \(q\) into the expense function \(E(q) = 20q + 100\).
For \(q = 5\):
\[E(5) = 20(5) + 100 = 100 + 100 = 200\]
For \(q = 10\):
\[E(10) = 20(10) + 100 = 200 + 100 = 300\]
For \(q = 25\):
\[E(25) = 20(25) + 100 = 500 + 100 = 600\]
Each step involves multiplying the quantity by the variable cost and then adding the fixed cost to determine the final total expense for that quantity.
- Final Results
Based on the calculations, we can complete the requested table to show the total expenses for the given quantities:
\[ \begin{array}{|l|l|} \ \text{Quantity } (q) & \text{Total Expense } (20q + 100) \\ 5 & 200 \\ 10 & 300 \\ 25 & 600 \\ \end{array} \]
The total expenses for quantities 5, 10, and 25 are 200, 300, and 600, respectively.
Examples
In real-world business, understanding fixed and variable costs is essential for setting prices and planning production. For example, a small bakery might have a fixed monthly rent of \$100 and a variable cost of \$20 to bake each batch of 50 cookies. By using the cost function \(E(q) = 20q + 100\), the bakery owner can predict total monthly expenses based on the number of batches \(q\) they plan to bake, allowing them to manage their budget and ensure the business remains profitable.