Problem 53 - A Graphical Approach to College Algebra

Question

(Modeling) In Exercises \(53-56\), assume that a linear relationship exists between the two quantities. 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) The cost function is \(y = 640x + 1100\). (b) 25 heaters cost \$17100. (c) Graph supports linear model.

Step 1: Identify Data Points

The two given data points from the problem are: 1. For 10 solar heaters, the cost is \(7500, giving us the point \((x_1, y_1) = (10, 7500)\).2. For 20 solar heaters, the cost is \)13900, giving us the point \((x_2, y_2) = (20, 13900)\).These points will be used to determine the linear function.

Step 2: Calculate the Slope

The slope \(m\) of the linear function is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the values: \[ m = \frac{13900 - 7500}{20 - 10} = \frac{6400}{10} = 640 \] Thus, the slope \(m\) is 640.

Step 3: Write the Equation of the Line

Using the point-slope form of the line \(y - y_1 = m(x - x_1)\) and substituting \(m = 640\) and the point \((10, 7500)\), we have:\[ y - 7500 = 640(x - 10) \] Solving for \(y\):\[ y = 640x - 6400 + 7500 \] \[ y = 640x + 1100 \] Hence, the cost function is \(y = 640x + 1100\).

Step 4: Calculate Cost for 25 Heaters

Substitute \(x = 25\) into the cost function \(y = 640x + 1100\):\[ y = 640(25) + 1100 \]\[ y = 16000 + 1100 = 17100 \]Therefore, the cost to produce 25 heaters is \$17100.

Step 5: Validate with Graphical Representation

To support the analytical results graphically, plot the line \(y = 640x + 1100\) and mark the points (10, 7500), (20, 13900), and compute (25, 17100). The linearity of the plot and alignment with the points confirm calculations.

Key Concepts

Slope Calculation

To find the slope of a linear function, we need two data points. In our case, these points are (10, 7500) and (20, 13900), representing the cost of producing solar heaters. The slope is essentially the change in cost (y-values) divided by the change in the number of heaters (x-values). The formula for calculating the slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

For our example:

• \( m = \frac{13900 - 7500}{20 - 10} = \frac{6400}{10} = 640 \)

This indicates that for each additional heater produced, the cost increases by $640. The slope helps us understand the rate at which the cost grows relative to the number of heaters.

Point-Slope Form

The point-slope form is a method used to write the equation of a line using a point on the line and the line's slope. The general form is:

• \( y - y_1 = m(x - x_1) \)

where \( (x_1, y_1) \) is a given point, and \( m \) is the slope.In our example, using the slope derived as 640 and the point (10, 7500), the equation becomes:

• \( y - 7500 = 640(x - 10) \)

After simplifying, you get:

• \( y = 640x + 1100 \)

This is our cost function, which accurately represents the linear relationship between the number of heaters and their production cost.

Cost Function

A cost function is a mathematical expression that calculates the total cost based on the number of items produced. It offers valuable insights into how costs change with production levels.In the context of our linear equation, the cost function is given by:

• \( y = 640x + 1100 \)

Here:

• \( y \) is the total cost,
• \( x \) represents the number of heaters,
• 640 is the cost per heater,
• 1100 is the fixed cost when no heaters are produced.

Using this function, one can easily predict the cost for any number of heaters by substituting the desired quantity into \( x \). For instance, the cost of 25 heaters is calculated by setting \( x = 25 \), yielding \( y = 17100 \).

Graphical Representation

Graphical representation is a visual way to validate and understand the relationship expressed by a linear equation. By plotting the function, you can confirm if your calculations align correctly with the data points.For our situation, the cost function \( y = 640x + 1100 \) can be plotted on a graph:

• The x-axis represents the number of heaters,
• The y-axis shows the total cost.

Plotting the points (10, 7500), (20, 13900), and (25, 17100) on this line should confirm the linear relationship. If the points lie on the line defined by the function, it verifies that the slope and the equation have been calculated appropriately. Moreover, visually interpreting the slope helps in appreciating the rate of increase in costs per heater.