Problem 60 - Algebra A Combined Function

Question

Solve. See Example 7. A fruit company recently released a new applesauce. By the end of its first year, profits on this product amounted to \(\$ 30,000\). The anticipated profit for the end of the fourth year is \(\$ 66,000\). The ratio of change in time to change in profit is constant. Let \(x\) be years and \(y\) be profit. a. Write a linear equation that relates profit and time. [Hint: Use the ordered pairs (1,30,000) and \((4,66,000) .]\) b. Use this equation to predict the company's profit at the end of the seventh year. c. Predict when the profit should reach \(\$ 126,000\).

Step-by-Step Solution

Answer

a. \(y = 12,000x + 18,000\); b. $102,000; c. Ninth year.

Step 1: Identify the Ordered Pairs

In the problem, you are given two points: (1, 30,000) and (4, 66,000). These points represent the time in years and the associated profits at those times.

Step 2: Calculate the Slope

The slope (m) for a linear equation is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (1, 30,000) \) and \( (x_2, y_2) = (4, 66,000) \). Substitute these values in to get:\[ m = \frac{66,000 - 30,000}{4 - 1} = \frac{36,000}{3} = 12,000 \]

Step 3: Write the Linear Equation

Using the point-slope form \( y - y_1 = m(x - x_1) \) and substituting one of the points, let's use (1, 30,000), and the slope:\[ y - 30,000 = 12,000(x - 1) \]Simplify this to get the equation of the line:\[ y = 12,000x + 18,000 \]

Step 4: Predict the Profit at the End of Seventh Year

Substitute \(x = 7\) into the equation \( y = 12,000x + 18,000 \):\[ y = 12,000(7) + 18,000 = 84,000 + 18,000 = 102,000 \]The predicted profit at the end of the seventh year is $102,000.

Step 5: Determine When the Profit will be $126,000

Here, you're solving for \(x\) when \(y = 126,000\) using the equation \( y = 12,000x + 18,000 \):\[ 126,000 = 12,000x + 18,000 \]Subtract 18,000 from both sides:\[ 108,000 = 12,000x \]Divide by 12,000:\[ x = \frac{108,000}{12,000} = 9 \]The profit is predicted to reach $126,000 at the end of the ninth year.

Key Concepts

Understanding Slope Calculation

In the world of linear equations, the slope is an essential component. It tells us how fast or slow something changes and in which direction. Imagine the slope as the steepness of a hill. In math terms, it is calculated through the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( m \) represents the slope, while \( (x_1, y_1) \) and \( (x_2, y_2) \) are points on a line.

In our exercise, we have the points \( (1, 30000) \) and \( (4, 66000) \). By plugging these numbers into our formula, we find:

• \( m = \frac{66000 - 30000}{4 - 1} \)
• \( m = \frac{36000}{3} = 12000 \)

This result, \( 12000 \), tells us that the profit increases by \$12000 each year. Knowing how to calculate the slope helps us understand the rate of change in any situation.

Exploring the Point-Slope Form

The point-slope form of a linear equation gives us a straightforward way to create an equation knowing just a point on the line and the slope. It looks like this: \( y - y_1 = m(x - x_1) \).

For our exercise, we use the slope \( m = 12000 \) and the point \( (1, 30000) \). Substituting these into the formula, we find:

• \( y - 30000 = 12000(x - 1) \)

By simplifying, we have:

• \( y = 12000x + 18000 \)

This is our linear equation, linking time (in years) to profit. Whenever you have a point and a slope, this form becomes a very handy tool to create linear equations.

Profit Prediction with Linear Equations

Predicting future profits using a linear equation is like looking into a crystal ball, but with math. Our equation \( y = 12000x + 18000 \) is ready to make predictions.

To predict the profit at the end of the seventh year, substitute \( x = 7 \) into the equation:

• \( y = 12000(7) + 18000 = 102000 \)

This means the expected profit should be \\(102,000.

But what if you want to find when the profit hits \\)126,000? Set \( y = 126000 \) and solve for \( x \):

• \( 126000 = 12000x + 18000 \)
• \( 108000 = 12000x \)
• \( x = 9 \)

Hence, the profit should reach \$126,000 by the ninth year. These predictions make linear equations a powerful tool in forecasting future outcomes.