Other # 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 5000 shirts can be sold at a price of \$ 72 , while 10000 shirts can be sold at a price of \$ 42 . Give a linear equation in the form p = m n + b that gives the price p they can charge for n shirts.p =

p = -0.002n +145

You are given two points: (n, p) = (46000, 53) and (50000, 45). The 2-point form of the equation of a line is useful for this.

y = (y2 -y1)/(x2 -x1)(x -x1) +y1

p = (45 -53)/(50000 -46000)(n -46000) +53

p = -8/4000(n -46000) +53

p = -0.002n +145 . . . . . demand equation

n = (-1000/3)p + (97000)/3)

The data given says, in effect, that the linear equation based on price passes through 2 points (82, 5000) and (91, 2000), given two points on a line, we'll use the point-slope form to start.  To use this we must first find the slope

m = (y2 - y1) / (x2 - x1) = (2000 - 5000) / (91 - 82) = -3000 / 9 = -1000/3

(I like to leave slopes as fractions <rather than decimal> since fractions are more accurate)

point-slope form

y - y0 = m(x - x0) at some point (x0, y0) with slope m, we'll use (91, 2000)

y - 2000 = (-1000/3)(x - 91), distribute

y - 2000 = (-1000/3)x + 91000/3, add 2000 to each side

y = (-1000/3)x +97000/3

oops, substitute p for x and n for y

n = (-1000/3)p + (97000)/3)

p

We have the points (2000,91) and (5000,82)

Then

The equation that gives the price p they can charge for n shirts is

p = -0.005n + 82

Establish the variables for the equation

n = number of shirts that can sell

p = price per shirt

For case one we have n1 = 5000 p1 =\$57

For case two we have n2 = 15000 p2= \$7

Calculate the slope remeber that in this case the y will be represented by the price (p) and the x by the number (n) so we have:

m =

Replace in the equation (y-y1) = m (x - x1) with our variables:

(p-p1) = m(n-n1)

p - 57 = -0.005 (n - 5000)

p - 57 = -0.005n + 25

p = -0.005n + 25 + 57

p = -0.005n + 82

To verify we can replace for example the values of n2 to get p2 as follows

p= -0.005 (15000) + 82

p = - 75 + 82

p = 7

If fulfills the condition that for 15000 shirts the price is \$ then the equation is correct

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