Question

a clothing business finds 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 13000 shirts can be sold at a price of \\$14, while 15000 shirts can be sold at a price of \\$8. give a linear equation in the form (p = mn + b) that gives the price (p) they can charge for (n) shirts.

answer: (p = 0.003n - 25)

round the value of your slope to three decimal places. be careful to use the proper variable and use the preview button to check your syntax before you submit your answer.

Explanation:

⚡ Using what you learned: Creating Equations to Model Relationships

Step 1: Identify the data points

The relationship is linear between the number of shirts, \(n\), and the price, \(p\). This gives us coordinate pairs in the form \((n, p)\):

$$ (n_1, p_1) = (13000, 14) $$

$$ (n_2, p_2) = (15000, 8) $$

Step 2: Calculate the slope \(m\)

$$ m = \frac{p_2 - p_1}{n_2 - n_1} $$

$$ m = \frac{8 - 14}{15000 - 13000} $$

$$ m = \frac{-6}{2000} $$

$$ m = -0.003 $$

Step 3: Find the vertical intercept \(b\)

Using the slope-intercept form \(p = mn + b\) and the point \((13000, 14)\):

$$ 14 = -0.003(13000) + b $$

$$ 14 = -39 + b $$

$$ b = 53 $$

Step 4: Write the final equation

$$ p = -0.003n + 53 $$

Answer:

\(p = -0.003n + 53\)