SOLUTION: It costs a company $1,550 to produce 500 items. The revenue on these 500 items is $1,700. The profit on 700 items is $410. Find the cost function and the revenue function. {{{R(x)=

Question 1155167: It costs a company $1,550 to produce 500 items. The revenue on these 500 items is $1,700. The profit on 700 items is $410. Find the cost function and the revenue function. . How do you get the cost function? Thank you.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
r = revenue
c = cost
p = profit
p = r - c

you have r = 17x/5.
you also have two profit points.
they are (500,150), (700,410)
all equations are presumed to be linear in this problem.
first look at the profit equation.
it's a straight line whose general slope intercept form of the equation is y = mx + b.
m is the slope and b is the y intercept.
slope = (y2-y1) / (x2-x1)
set (x1,y1) = (500,150)
set (x2,y2) = 700,410)
slope becomes (410-150)/(700-500) = 260/200 = 26/20 = 13/10.
equation becomes y = 13/10 * x + b
replace x and y with one of the points to get:
150 = 13/10 * 500 + b
solve for b to get:
b = 150 - 13/10 * 500 + b
simplify to get b = -500
your profit equation becomes y = 13/10 * x - 500
replace y with p and you get p = 13/10 * x - 500

you know that p = r - c
p = 13/10 * x - 500
r = 17/5 * x
you get 13/10 * x - 500 = 17/5 * x - c
solve for c to get c = 17/5 * x - (13/10 * x - 500)
simplify to get c = 17/5 * x - 13/10 * x + 500
multiply 17/5 * 2/2 to get:
c = 34/10 * x - 13/10 * x + 500
combine like terms to get c = 21/10 * x + 500

you now have all three equations.
you have:
r = 17/5 * x
p = 13/10 * x - 500
c = 21/10 * x + 500

since p = r - c, then:
p = 13/10 * x - 500 and p = 17/5 * x - (21/10 * x + 500) should be equivalent.
if they are equivalent, they will draw the same line on a graph.
here's what the graph looks like.

$$$

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