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 11000 shirts can be sold at a price of $72, while 12000 shirts can be sold at a price of $67. Give a linear equation in the form p=mn+b that gives the prices they can charge
The correct answer and explanation is:
We are asked to find a linear equation in the form p=mn+bp = mn + b, where pp is the price per shirt and nn is the number of shirts sold. The given data points are:
- When 11,000 shirts are sold, the price per shirt is $72.
- When 12,000 shirts are sold, the price per shirt is $67.
Step 1: Define Variables
Let:
- nn represent the number of shirts sold.
- pp represent the price per shirt.
The equation we need to find has the form: p=mn+bp = mn + b
where mm is the slope of the line (the change in price per unit change in the number of shirts sold) and bb is the y-intercept (the price when no shirts are sold, though it might not be meaningful in the context of this scenario).
Step 2: Calculate the Slope (mm)
The slope of a line is calculated using the formula: m=p2−p1n2−n1m = \frac{{p_2 – p_1}}{{n_2 – n_1}}
where:
- (n1,p1)=(11000,72)(n_1, p_1) = (11000, 72) is the first data point.
- (n2,p2)=(12000,67)(n_2, p_2) = (12000, 67) is the second data point.
Substituting the values: m=67−7212000−11000=−51000=−0.005m = \frac{{67 – 72}}{{12000 – 11000}} = \frac{{-5}}{{1000}} = -0.005
So, the slope mm is −0.005-0.005. This means for every 1-shirt increase in sales, the price per shirt decreases by $0.005.
Step 3: Use the Point-Slope Form to Find the Equation
We now have the slope m=−0.005m = -0.005 and we can use one of the points to find the y-intercept bb. We will use the point (11000,72)(11000, 72).
Substitute n=11000n = 11000 and p=72p = 72 into the equation p=mn+bp = mn + b: 72=(−0.005)(11000)+b72 = (-0.005)(11000) + b 72=−55+b72 = -55 + b b=72+55=127b = 72 + 55 = 127
Step 4: Write the Final Equation
Now that we have both the slope and the y-intercept, the equation becomes: p=−0.005n+127p = -0.005n + 127
Interpretation
This equation tells us that for every 1-shirt increase in sales, the price per shirt decreases by $0.005. When 11,000 shirts are sold, the price is $72, and when 12,000 shirts are sold, the price is $67. This equation helps the business predict prices based on the number of shirts sold.