Problem 116 - College Algebra

Question

A clothing business fi ds there is a linear relationship between the number of shirts, \(n,\) it can sell and the price, \(p\), it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of \(\$ 30,\) while 3,000 shirts can be sold at a price of \(\$ 22 .\) Find a linear equation in the form \(p(n)=m n+b\) that gives the price \(p\) they can charge for \(n\) shirts.

Step-by-Step Solution

Answer

The linear equation is: \(p = -0.004n + 34\).

Step 1: Identify the Points

We have two points based on the historical data: - Point 1: \((n_1, p_1) = (1000, 30)\)- Point 2: \((n_2, p_2) = (3000, 22)\). These points will be used to calculate the slope \(m\) of the linear equation.

Step 2: Find the Slope

The slope \(m\) of the line is given by the formula \(m = \frac{p_2 - p_1}{n_2 - n_1}\). Substituting the values:\[m = \frac{22 - 30}{3000 - 1000} = \frac{-8}{2000} = -0.004\]. The slope is \(-0.004\).

Step 3: Write the Equation in Point-Slope Form

Using the point-slope form of a line, which is \(p - p_1 = m(n - n_1)\), and using Point 1 \((1000, 30)\), we substitute in the values:\[p - 30 = -0.004(n - 1000)\].

Step 4: Simplify to Find the Linear Equation

Solve the point-slope form to get the linear equation:1. Distribute \(-0.004\) through the equation: \[p - 30 = -0.004n + 4\].2. Solve for \(p\): \[p = -0.004n + 4 + 30\].3. Simplify:\[p = -0.004n + 34\]. This is the linear equation relating the number of shirts to the price per shirt.

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. In our exercise, we have a linear relationship between the price charged per shirt and the number of shirts sold. The slope helps us understand how much the price changes for each additional shirt sold.
The calculation of the slope, denoted by the letter \( m \), is done using two points from the data. In this case, the points are \((1000, 30)\) and \((3000, 22)\). The formula for the slope is:

• \( m = \frac{p_2 - p_1}{n_2 - n_1} \)

Substituting the given values:

• \( m = \frac{22 - 30}{3000 - 1000} = \frac{-8}{2000} = -0.004 \)

This negative slope tells us that as the price decreases, the number of shirts sold increases.

Point-Slope Form

Point-slope form is a way to express the equation of a line using a specific point on the line and its slope. This form is particularly useful when you already know one point and the slope, as it allows you to easily derive the equation.To find the equation in point-slope form, we use the formula:

• \( p - p_1 = m(n - n_1) \)

Here, \( (n_1, p_1) \) is one of our points, \((1000, 30)\), and \( m \) is the slope, which we calculated as \(-0.004\). Plugging these values into the formula:

• \( p - 30 = -0.004(n - 1000) \)

The point-slope form is a stepping stone to obtaining the standard equation of the line.

Equation of a Line

The equation of a line represents the relationship between two variables. In our exercise, the equation shows how the price changes with the number of shirts sold. The general form of a line's equation is \( p(n) = m \cdot n + b \).To find the equation for our scenario, we start from the point-slope form:

• \( p - 30 = -0.004(n - 1000) \)

By simplifying, we distribute and rearrange terms to find:

• \( p = -0.004n + 4 + 30 \)
• \( p = -0.004n + 34 \)

This is the linear equation that expresses the price \( p \) they can charge for \( n \) shirts. It shows a clear mathematical relationship between price and quantity.

Pricing Strategy

A pricing strategy in business involves decisions on how to set the price levels for products or services. With linear equations, businesses like the clothing company in our exercise can predict how changes in price might affect sales volume.
By using historic data and deriving a linear model, the company can identify the rate at which price changes affect the quantity sold. In this context, the negative slope \(-0.004\) suggests a strategic adjustment: as the price decreases by \$1, the company can expect to sell approximately 250 more shirts.

• This understanding allows for strategic decision-making.
• Businesses can balance between pricing and quantity sold for optimal revenue.

Therefore, using the linear equation \( p = -0.004n + 34 \), the business can develop structured plans for setting competitive prices while maintaining profitable sales volumes.