Originally Posted by

**Rob** Hi, I have been trying to work on this problem and can't seem to figure it out. Please help me with this problem. Thanks

Problem:

The cost to produce 50 hard-shell camera cases is $1500, and the cost to produce 80 of the cases is $1800. Each of the cases sells for $40. If all the cases produced can be sold, and that both cost and revenue are given by linear functions, find the cost function, the

revenue function, graph them and show the break-even point, and

(with work shown) find the break-even point.

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