Originally Posted by Rob
We are told that these are linear functions, so we know they will be of the form: y = mx + b, where m is the slope and b is the y-intercept.
Let C(x) be the cost function and R(x) be the revenue function, where x is the number of units sold.
Revenue is the money recieved from selling x units, that is, the revenue is 40x, since we get 40 dollars for each unit we sell. So:
R(x) = 40x
Finding C(x) is a bit more challenging. First we need the slope. we are told the cost to produce 50 hard-shell camera cases is $1500, and the cost to produce 80 of the cases is $1800.
That is C(50) = 1500 and C(80) = 1800
=> (x1,y1) = (50, 1500) and (x2,y2) = (80, 1800)
using the slope formula:
m = (y2 - y1)/(x2 - x1) = (1800 - 1500)/(80 - 50) = 300/30 = 10
now we use either of the points and our calculated value for m and plug them into the point-slope form:
Using (x1,y1) = (50,1500), m = 10
y - y1 = m(x - x1)
=> y - 1500 = (10)(x - 50)
=> y = 10x - 500 + 1500
=> y = 10x + 1000
=> C(x) = 10x + 1000
The break even point is where the revenue equals the cost of production, that is, where:
10x + 1000 = 40x
=> 30x = 1000
=> x = 33 1/3
so after 33 1/3 units are sold, we break even