Problem 57 - Algebra A Combined Function

Question

Solve. See Example 8. The cost in dollars \(y\) of producing \(x\) computer desks is given by \(y=80 x+5000\). a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 100 & 200 & 300 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the number of computer desks that can be produced for \(\$ 8600 .\) (Hint: Find \(x\) when \(y=8600 .)\)

Step-by-Step Solution

Answer

a. The table is complete with \( y \) values of 13000, 21000, and 29000. b. 45 desks can be produced for \( \$8600 \).

Step 1: Understanding the Equation

The equation given is \( y = 80x + 5000 \). In this equation, \( y \) represents the cost in dollars, and \( x \) represents the number of computer desks produced. The cost function is linear with a fixed cost of \( 5000 \) dollars plus \( 80 \) dollars per desk.

Step 2: Calculating Costs for the Table

We need to use the equation \( y = 80x + 5000 \) to find \( y \) for each given value of \( x \). - For \( x = 100 \): \[ y = 80(100) + 5000 = 8000 + 5000 = 13000 \] - For \( x = 200 \): \[ y = 80(200) + 5000 = 16000 + 5000 = 21000 \] - For \( x = 300 \): \[ y = 80(300) + 5000 = 24000 + 5000 = 29000 \]

Step 3: Completing the Table

Now we fill in the table with the calculated \( y \) values:\[\begin{array}{|c|c|c|c|}\hline \boldsymbol{x} & 100 & 200 & 300 \hline \boldsymbol{y} & 13000 & 21000 & 29000 \hline \end{array}\]

Step 4: Finding Desks for a Given Cost

We need to find \( x \) when \( y = 8600 \). Using the equation \( y = 80x + 5000 \), substitute \( y = 8600 \) and solve for \( x \): \[ 8600 = 80x + 5000 \] Subtract 5000 from both sides: \[ 3600 = 80x \] Divide by 80: \[ x = \frac{3600}{80} = 45 \]

Step 5: Conclusion

The table has been completed with the correct cost values for \( x = 100, 200, 300 \). Additionally, the number of desks that can be produced for \( \$8600 \) is \( 45 \).

Key Concepts

Understanding the Cost Function

A cost function is a mathematical expression that describes how production costs vary with the quantity of output. In this scenario, the cost function is given by the equation \( y = 80x + 5000 \). Here, \( y \) represents the cost in dollars to produce \( x \) number of computer desks. The formula shows two components: a fixed cost, which is the constant 5000, and a variable cost, which depends on the number of desks produced. - **Fixed Cost:** The 5000 dollars is a fixed cost. This is the baseline cost of production, regardless of how many desks are made. It might include things like rent, utilities, and salaries. - **Variable Cost:** The term 80x represents the variable cost. For each desk produced, an additional $80 is added to the overall cost. This could cover materials, additional labor, or other costs that increase with production.Understanding cost functions helps businesses manage their budgeting and pricing strategies efficiently.

Solving Linear Equations in Context

Solving linear equations involves finding the value of the variable that makes the equation true. In this context, once we have a cost function like \( y = 80x + 5000 \), we can solve it to find either \( x \) (number of desks) or \( y \) (total cost). Let's explore how it's done:- **Finding Cost for a Given Number of Desks:** Suppose you need to calculate the production cost for a specific number of desks, say 100, 200, or 300. Substitute these values of \( x \) into the cost equation: - For 100 desks: \( y = 80(100) + 5000 \) - For 200 desks: \( y = 80(200) + 5000 \) - For 300 desks: \( y = 80(300) + 5000 \) - **Finding Number of Desks for a Given Cost:** In another scenario, you may need to determine how many desks can be made with a fixed budget of, say, $8600. You set \( y = 8600 \) and rearrange the equation to solve for \( x \): - Start by subtracting 5000 from both sides to isolate the term involving \( x \). - Then, divide by 80 to solve for \( x \). Thus, mastering these steps can help solve many real-life budgeting problems in production.

Using a Table of Values for Clarity

Creating a table of values helps in visualizing the relationship between two quantities, such as the number of desks produced and the total production cost. A table can quickly show how changes in one variable affect the other.- **Tables for Data Representation:** When you have a formula like \( y = 80x + 5000 \), constructing a table is straightforward. Input various values of \( x \) (such as 100, 200, 300) and calculate the corresponding \( y \) values using the formula. - **How It Helps:** This visual representation can make patterns more evident and data comparison easier. You can immediately see that as \( x \) increases, \( y \) also increases, ensuring that the relationship is linear.Using tables is especially helpful in preliminary studies where you gather results for different scenarios and need an organized way to compare them. By keeping values aligned, tables remove ambiguity and clearly display the relationship between the inputs and outputs.