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0:00 in this video we're going to talk about how to write the cost function the revenue function the profit function as well as how to find the break even point so let's dive in we've got a company that manufactures teddy bears for $150 each and it has equipment costs of $2,500 each teddy bear sells for $4 each can you write a cost function a revenue function a profit function and then can you find the break even point so let's take this one step at a time now some equations that you might want to write down or memorize are the basic slope intercept form of the equation of a line as well as your profit equation which remember profit is equal to the revenue the money that you
0:46 take in minus the cost the expenses or what you spend going out so that difference is your leftover which is your profit right so let's break this down one step at a time so for the cost let's say C of X or X is the number of teddy bears sold it's going to cost $1.50 times x the number of teddy bears that you're making or manufacturing plus it also costs you the initial expense of the $2,500 for the equipment so remember when you think of the slope or the M value that's like the rate like how much it costs per teddy bear to make whereas the B that's our Y intercept or our initial condition or our starting amount and these are our starting expenses the 2,500 that's just a fixed or constant
1:33 cost whereas this is going to increase as you make more and more teddy bears so that's our cost function now the revenue function that's how much money that you're making right when you sell these teddy bears and they're saying that we sell them for $4 each so it's going to be $4 times the number of teddy bears that you sell if you sell 10 it's going to be four time 1040 $40 that you're taking in and then our profit is again like we said the difference so we're going to say p ofx is equal to R ofx our Revenue minus our expenses or our costs now one mistake that students sometimes make is they forget to use parentheses so our revenue is 4X our cost is this whole thing so let's put it in parentheses let's treat it like a
2:21 group and now what we're going to do is we're just going to distribute this negative -1 into the parentheses which is going to make this a1. 5x minus 2500 and if we combine like terms let's see 4x - 1.5x is 2.5x minus 2,500 and that's our profit okay as a function of the number of teddy bears okay X now to find the break even Point what is the break even well basically the break even is when the amount of money that you've taken in is the exact same amount as the cost for your expenses going out so really that profit at that break even point is going to be zero after which then you start to
3:08 become profitable so what we're going to do is we're going to set this P of X here equal to zero and we're going to solve so let's go ahead and do that we've got zero equal 2.5x minus 2,500 I'm going to add the 2500 to both sides to get the variable by itself okay and so you can see we have 2,500 is equal to 2.5x we're trying to solve for the number of teddy bears that need to be uh sold and made uh in order to make uh a profit of zero that's our break even so divide both sides by 2.5 and 2,500 divided 2.5 is equal to
3:53 1,000 teddy bears and that would be your break even point so once you sell th1 then you're into the uh black uh as far as being profitable and if you sell less than this you're in the red whereas you're uh still in the you know basically you haven't made money yet you haven't broken even so great job if you want to see another example where I talk about profit Revenue cost I'll put another video right there follow me over to that video and try maybe some of those problems on your own for extra practice I'll see you in that video
