# Linear Function of a company's cell phone production

A company manufactures cell phones and sells them for $\$150$each. The fixed costs of producing the cell phones are$\$56,000$ plus a variable cost of $\$45$for each cell phone produced. Let$x$be the number of cell phones produced.$(1)$Let$C(x)$be the cost function. Find$C(x)$.$(2)$Let$R(x)$be the revenue function. Find$R(x)$.$(3)$Let$P(x)$be the profit function. Find$P(x)$in terms of$x(4)$What is the profit if$50,000$cell phones are sold?$(5)$Find the break even point. I'm sort of confused on how to set up the functions, and what is a break even point? What do they mean by "variable cost" and "fixed cost". Any sort of help would be great. Including answers! ## 2 Answers Here's my try. (1) I suppose$\$56,000$ is a fixed cost (fabric, salaries etc), so if we wanna find the cost function it'll be as a constant cause it doesn't depend on the number of phones sold. How ever $\$45$is the cost of producing one cell, thus it should be the variable's coefficient (I guess this cost involves cost such as the cell phone's material)$C(x)= 45x + 56000$(2) This function should represent all the income, which will depend on the number of cell sold:$R(x)=150x$(3) Profit is the difference between the cost and the revenue:$P(x)=R(x)-C(x)=150x - 45x - 56000= 105x - 56000$(4) Profit of$50 ,000$cell phones sold:$P(50000)=105*50000 - 56000$(5) I'm not sure what break even point means, but I guess it's the point where the profits are equal to the cost, so they want you to find this point:$105*x-56000=0105*x=56000x=56000/105=533,333$Thus if you sell$533\$ cell phones you won't have any profit, because you'll spend all the income in basic costs (fixed costs).

I am surprised that you would be given a problem like this without being told what those words mean. The "fixed costs" are costs you have to pay no matter how many you make- renting the work space, paying for electricity, water and sewage, etc, that you have to pay even if there is no one working. The "variable costs" are costs that you have to pay for each one produced- that might be the cost of parts, the wages of the person assembling it, etc. The total cost of producing x items will be those "fixed costs" plus the "variable costs" times x. Here that is 5600+ 45x. The profit is the amount of money brought in, 150x minus the cost: 150x- (5600+ 45x)= 105x- 5600. If that is negative, you are losing money. If that is positive, you are earning money. You "break even" if you neither lose nor earn money: if your profit, 105x- 5600= 0.