Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Solar Heater Production A company produces 10 solar heaters for \(\$ 7500 .\) The cost to produce 20 heaters is \(\$ 13,900\) (a) Express the cost \(y\) as a linear function of the number of heaters, \(x\) (b) Determine analytically the cost to produce 25 heaters. (c) Support the result of part (b) graphically.

Step-by-Step Solution

Answer

(a) Cost function: \( y = 640x + 1100 \) (b) Cost for 25 heaters: \( \$17100 \) (c) Graph verifies the calculation.

Step 1: Calculate the Slope

To express the cost as a linear function, we need to calculate the slope (m) using the given points (10, 7500) and (20, 13900). The formula for the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the values, we get: \[m = \frac{13900 - 7500}{20 - 10} = \frac{6400}{10} = 640.\]

Step 2: Find the y-intercept

Next, use the slope-point form to find the y-intercept (b). The formula is \( y = mx + b \). Using point (10, 7500), we have: \[7500 = 640 \times 10 + b.\]Solving for b, we get:\[7500 = 6400 + b \Rightarrow b = 7500 - 6400 = 1100.\]

Step 3: Write the Linear Function

Now with the slope and the y-intercept, we can write the cost function as:\[y = 640x + 1100.\]

Step 4: Calculate the Cost for 25 Heaters

Using the linear function \( y = 640x + 1100 \), substitute \( x = 25 \) to find the cost.\[y = 640 \times 25 + 1100 = 16000 + 1100 = 17100.\]So, the cost to produce 25 heaters is \( \$17100 \).

Step 5: Support Graphically

To graphically support the result, plot the linear equation \( y = 640x + 1100 \) on a graph. Plot the points (10, 7500) and (20, 13900) from which the line passes. Extend the line to \( x = 25 \) and verify that \( y = 17100 \) matches our calculated cost.

Key Concepts

Slope Calculation

The slope is a measure of how steep a line is. It tells us how much the `y` value (or cost, in this case) changes for every one unit increase in `x` (the number of heaters). To find it, we use two known points on the line,

• (10, 7500) and
• (20, 13900),

applying the formula: \[\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}.\]

When we plug in those points: \[m = \frac{13900 - 7500}{20 - 10} = \frac{6400}{10} = 640.\]

This tells us that for each additional solar heater produced, the cost increases by $640.

Y-intercept

The y-intercept is the point where the line crosses the `y`-axis. It's the starting value of the cost when no heaters are produced.

By using the slope-point form, we can find the y-intercept `b` of our linear function: \[y = mx + b.\]

Taking the point (10, 7500), we have: \[7500 = 640 \times 10 + b.\]

Solving for `b` gives: \[7500 = 6400 + b \Rightarrow b = 7500 - 6400 = 1100.\]

Thus, the y-intercept is 1100. This means that even before producing any heaters, there is a base cost of $1100.

Graphical Representation

Visualizing the linear function helps us better understand how costs behave as production scales. We can represent the function \[y = 640x + 1100.\] on a graph by plotting:

• The initial points (10, 7500) and (20, 13900).
• Extend the line to verify it passes through calculated costs like the cost for 25 heaters.

You can see that the line is straight, indicating a consistent cost increase per heater due to the constant slope. When plotting for `x = 25`, check that it aligns with \[y = 17100.\] This visual method supports our calculated outcome, affirming the accuracy of the linear model.

Cost Analysis

Understanding how costs change with production is critical for businesses. Our linear function \[y = 640x + 1100\] provides a simple, yet powerful tool for analysis.

This equation can help:

• Predict costs for different levels of production.
• Identify fixed costs (y-intercept) and variable costs (slope).
• Plan budgets by predicting how total costs will rise with increased output.

For instance, the cost for producing 25 heaters was calculated using this formula, resulting in $17100. Businesses can rely on such an analysis for planning and strategizing efficiently.