Problem 79
Question
The production cost, in dollars, for \(x\) color brochures is \(C(x)=500+3 x .\) The fixed cost is \(\$ 500\) since that is the amount of money needed to start production even if no brochures are printed. (a) If the fixed cost is decreased by \(\$ 50,\) find the new cost function. (b) Graph both cost functions and interpret the effect of the decreased fixed cost.
Step-by-Step Solution
Verified Answer
a) The new cost function would be \(C'(x)=450+3x\). b) The graph would contain two lines representing the initial and new cost functions. The line representing the new function starts at a lower point due to the reduced fixed cost, but would run parallel to the original line, as the cost per brochure remains the same.
1Step 1: Formulate new cost function
The new cost function after the decrease of $50 in the fixed cost would be a modification of the original function \(C(x)=500+3x\). The modified function \(C'(x)\) would look like: \(C'(x)=(500-50)+3x=450+3x\)
2Step 2: Graph the original cost function
First, graph the original cost function \(C(x)=500+3x\) on a coordinate plane. Plot the y-intercept (0,500), which represents the fixed cost of $500. The slope of the function, which is 3, indicates that for each additional brochure printed (increase in x), the cost increases by $3. Connect these points to create a line.
3Step 3: Graph the new cost function
Now, graph the modified cost function \(C'(x)=450+3x\). The y-intercept is now (0,450), which denotes the new lowered fixed cost of $450. The slope still remains 3, meaning for each additional brochure printed, the cost increases by $3. Connect these points to create a second line on the same graph.
4Step 4: Analyze and interpret the graphs
A thorough look at both lines on the graph shows that the line for the modified function \(C'(x)=450+3x\) starts at a lower point on the y-axis, which represents the lowered fixed cost. Both lines have the same slope, which implies that despite the lowered fixed cost, the variable cost per brochure has not changed. Overall, the decreased fixed cost has lowered the total cost for any given number of brochures.
Key Concepts
Understanding Fixed and Variable Costs in Business MathematicsGraphing Linear Functions to Visualize CostInterpreting Graphs in Algebra for Cost Analysis
Understanding Fixed and Variable Costs in Business Mathematics
In business mathematics, a firm's total costs are typically divided into two categories: fixed costs and variable costs. Understanding these two components is crucial for financial planning and analysis.
Fixed costs are expenses that do not change with the level of goods or services produced by the business. These are costs that a company incurs even when it produces nothing at all. Common examples include rent, salaries, and insurance. In the given exercise, the fixed cost of producing color brochures was initially \(500, representing the baseline cost before any brochures are printed.
Variable costs, on the other hand, change with the level of output. They tend to increase with each additional unit of production. In our exercise, the variable cost is \)3 for every brochure printed. This reflects materials, labor, and other costs that increase directly with the number of brochures produced.
When we look at reducing fixed costs, as shown in the exercise where the fixed cost drops by \(50, the new fixed cost becomes \)450. This decrease lowers the overall cost function, translating into cost savings for the business.
Fixed costs are expenses that do not change with the level of goods or services produced by the business. These are costs that a company incurs even when it produces nothing at all. Common examples include rent, salaries, and insurance. In the given exercise, the fixed cost of producing color brochures was initially \(500, representing the baseline cost before any brochures are printed.
Variable costs, on the other hand, change with the level of output. They tend to increase with each additional unit of production. In our exercise, the variable cost is \)3 for every brochure printed. This reflects materials, labor, and other costs that increase directly with the number of brochures produced.
When we look at reducing fixed costs, as shown in the exercise where the fixed cost drops by \(50, the new fixed cost becomes \)450. This decrease lowers the overall cost function, translating into cost savings for the business.
Graphing Linear Functions to Visualize Cost
Graphing linear functions is a fundamental skill in algebra that helps visualize relationships between variables. In the context of cost functions, a linear graph aids in easily understanding how costs change as production varies.
To graph a linear function like the cost function for producing brochures, \(C(x) = 500 + 3x\), we start by identifying two key components: the y-intercept and the slope. The y-intercept is where the line crosses the y-axis, which corresponds to the fixed costs, and in this case is \( (0, 500) \). The slope indicates the rate of change, which here is 3, showing the incremental increase in cost per additional brochure.
When graphing, we plot the y-intercept and use the slope to determine how the line rises. A straight line is then drawn to represent the cost function. With the new cost function after a decrease in fixed costs, \(C'(x) = 450 + 3x\), we follow the same process but start at a lower y-intercept of \( (0, 450) \).
To graph a linear function like the cost function for producing brochures, \(C(x) = 500 + 3x\), we start by identifying two key components: the y-intercept and the slope. The y-intercept is where the line crosses the y-axis, which corresponds to the fixed costs, and in this case is \( (0, 500) \). The slope indicates the rate of change, which here is 3, showing the incremental increase in cost per additional brochure.
When graphing, we plot the y-intercept and use the slope to determine how the line rises. A straight line is then drawn to represent the cost function. With the new cost function after a decrease in fixed costs, \(C'(x) = 450 + 3x\), we follow the same process but start at a lower y-intercept of \( (0, 450) \).
Interpreting Graphs in Algebra for Cost Analysis
Interpreting graphs is an essential part of algebra, allowing us to draw meaningful conclusions from visual data. In our exercise, comparing the graphs of two cost functions illuminates the impact of changing fixed costs on total expenses.
Upon graphing the original and new cost functions, we note that both lines have the same slope—indicating that variable costs per brochure remain constant. However, the line representing the new cost function starts at a lower y-intercept (\(C'(x)\) starts at \( (0, 450) \)), showing the decreased fixed costs.
The effect of lower fixed costs on the total cost is clear: for any given number of brochures, the total cost is now less than before, as reflected by the lower starting point of the new cost function. This demonstrates an important principle in economics—the ability to reduce fixed costs can significantly impact overall profitability. By looking at the distance between the two parallel lines at any level of production \(x\), we can quantify the savings resulting from the reduced fixed cost.
Upon graphing the original and new cost functions, we note that both lines have the same slope—indicating that variable costs per brochure remain constant. However, the line representing the new cost function starts at a lower y-intercept (\(C'(x)\) starts at \( (0, 450) \)), showing the decreased fixed costs.
The effect of lower fixed costs on the total cost is clear: for any given number of brochures, the total cost is now less than before, as reflected by the lower starting point of the new cost function. This demonstrates an important principle in economics—the ability to reduce fixed costs can significantly impact overall profitability. By looking at the distance between the two parallel lines at any level of production \(x\), we can quantify the savings resulting from the reduced fixed cost.
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