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 1000 shirts can be sold at a price of $22, while 3000 shirts can be sold at a price of $6. Give a linear equation equation in the form p=mn+b that gives the price p they can charge for n shirts.

Answer 1

Answer:

Step-by-step explanation:

Let's start by making some x, y coordinates from our data.  n is the number of shirts sold (our x) and p is the price for n shirts (our y).  If we sell 1000 shirts at $22 the coordinate is (1000, 22).  Likewise, if we sell 3000 shirts at $6 the coordinate is (3000, 6).  The slope of the line is found from these 2 points:

$m=\frac{6-22}{3000-1000}=-\frac{3}{1000}$

Interpreting this slope is just as important as being able to find it.  Taking it within the context of our given data, this slope value of -3/1000 means that for every 1000 shirts we will we are losing $3.

We can write a linear equation from the data by picking one of the points and using these coordinates in the point-slope form of a line: y - y1 = m(x - x1) where x1 and y1 are the coordinates from the point we choose.  Let's use the coordinate (3000, 6).  Rest assured that you will get the same equation regardless of what point you choose.  Promise!!!!!!

$y-6=-\frac{3}{1000}(x-3000)$

which simplifies to

$y=-\frac{3}{1000}x+15, x\geq 1$

That means that you are basing your price on selling at least 1 shirt.  In the context of this problem, our starting price for 1 shirt is found by filling in a 1 for x, since x is the number of shirts.

f(1) = $-\frac{3}{1000}(1)+15$

which gives us a starting price of $14.997 or $15 rounded.  In terms of n and p,

$p=-\frac{3}{1000}n+15, n\geq 1$

That's the linear cost function for the shirts.