Assuming the relationship between price and sales is linear, find the linear equation that models the number of bikes sold based on price.

Question

A manufacturer of high-end bikes was selling 12,500 of their most popular bike model every month at a retail price of $1,300 for each bike. They raised the retail price to $1,500 and the number of bikes sold every month fell to 11,100. Assuming the relationship between price and sales is linear, find the equation that models the number of bikes sold based on price. If the manufacturer needs to charge more than $1,300 per bike, what should the price be in order to sell 12,000 bikes each month (round to 2 decimal places)?

Answer 1

As the relationship is defined as linear, we can use the slope-intercept equation: y = mx + b where m is the slope (chg in y (units) vs. chg in x (price)) and b is the y-intercept.

Slope is Δy/Δx = (11100 - 12500)/(1500 - 1300) = -1400/200 = -7

Can solve for b by substituting one of the pairs: 12500 = -7(1300) + b -> b = 21600

y = -7x + 21600 is the model.

To sell 12,000, substitute for y and solve for x: 12000 = -7x + 21600

You should get $1,371.43 rounded to 2 decimal places.

Answer 2

Put number sold on the y axis. Put price on the x axis.

It takes two points to identify a straight line. You have (1300, 12500) and (1500, 11100).

Find slope (m) and y-intercept (b).

slope = m =(y2-y1)/(x2-x1) = (11100-12500)/(1500-1300) = -1400/200 = -7

y = mx +b (substitute either point)

12500 = (-7)(1300) + b

b = 12500 + 9100 = 21600

The required equation is y = -7x + 21600 or #sales = -7(price) + 21600

Sell 12000:

12000 = -7x + 21600

x=(21600-12000)/7 = $1371.43