Question 1037039: A company’s manager estimated that the cost C, in dollars, of producing n items is C equals 7 n plus 350. The company sells each item for $12. The company makes a profit when total income from selling a quantity of items is greater than the total cost of producing that quantity of items. Which of the following inequalities gives all possible values of n for which the manager estimates that the company will make a profit?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let's see if we can find the right equation for this.
cost = 7 * number of units plus 350.
if we let c = the cost and n = the number of units, then the formula becomes c = 7*n + 350.
revenue - 12 * number of units.
if we let r - the revenue and n = the number of units, then the formula becomes r = 12 * n.
profit = revenue minus cost.
if we let p = profit and r = revenue and c = cost, then the formula becomes p = r - c.
we have 3 formulas:
c = 7 * n + 350 = cost formula.
r = 12 * n = revenue formula.
p = r - c = profit formula.
since we know that c = 7 * n + 350 and r = 12 * n, then we can substitute in the profit formula to get:
p = r - c becomes p = 12 * n - (7 * n + 350).
simplify this to get p = 12 * n - 7 * n - 350.
combine like terms to get p = 5 * n - 350.
in order for the company to make a profit, p has to be greater than 0.
the formula for profit becomes p > 0.
since p = 5 * n - 350, then you get 5 * n - 350 > 0.
solve for n in this formula to get n > 350 / 5 becomes n > 70.
the number of units sold has to be greater than 70 in order for the company to make a profit.
let's see what happens to each of these formulas.
if 70 units are sold, then you get.
cost = 7 * 70 + 350 = 840
revenue = 12 * 70 = 840
profit = revenue minus cost = 840 - 840 = 0.
0 profit is made when 70 units are sold.
you can see that, when less than 70 units are sold, the profit is negative which means the company takes a loss.
you can see that, when more than 70 units are sold, the profit is positive which means the company makes a profit.
based on the given equations, the break even point is 70 units.
the company makes a profit when more than 70 units are sold.
this can also be shown graphically.
you can graph the revenue equation and the cost equation and see that the revenue equation becomes greater than the cost equation when x is greater than 70.
that graph is shown below.
you can also graph the revenue equation minus the cost equation and see that the profit becomes greater than 0 when x is greater than 70.
that graph is shown below.
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