Business Question

Question

Now that sales are booming, it’s time to talk about shipping your action figures to more stores. One of your business managers gathered some data about your current shipping operations. Two of these data points are a 12-mile delivery that costs $11.75, and a 27- mile delivery that costs $19.23. Usethese data points to fit a linear function in slope-intercept form: 𝑦 = 𝑚𝑥 + 𝑏, where 𝑥 is the distance covered and 𝑦 is the cost. What are the values of 𝑚 and 𝑏? What do they represent?

a. Write the equation 𝑦=𝑚𝑥+𝑏, and plug in your data points for 𝑥 and 𝑦.

b. Solve the two resulting equations to find the unknowns 𝑚 and 𝑏. What do these values represent?

Answer 1

I will use distance (in miles) on the x axis, and cost (in $) on the y axis.

The two data points given are (12, 11.75) and (27, 19.23)

A linear function has a general equation y = mx + b as stated, where m is the gradient, and b is the y intercept.

The gradient we can see is positive as it goes up and to the right (i.e. price increases as distance increases).

Gradient = distance up / distance across

We have two points, so we can use the differences, so

m = (19.23 - 11.75) / (27 - 12) = 7.48 / 15 = 0.498666.. or 0.50 to 2 decimal places

So now our equation is y = (7.48/15) x + b [use the actual value of m for accuracy in calculating b]

We can now plug in the values of either of the points to find b

11.75 = (7.48/15) (12) + b

=> 11.75 = 7.48 x 4/5 + b

=> b = 11.75 - 7.48 x 4/5

=> b = 5.766 or 5.77 to 2 decimal places

Hence the equation is y = 0.50x + 5.77 OR 2y = x + 11.54

(or perhaps you can approximate with y = 1/2 x + 6 OR 2y = x + 12)

b ($5.77) represents a fixed cost, a cost even if you go 0 miles.

m ($0.50) represents the incremental cost for each additional mile.

Hence, for, say, 50 miles, the equation predicts a cost of $30.77 ($31 with the approximation).

Always good to double check, so for 12 miles, what's my cost? $5.77 + $6 = $11.77