Finding the break-even point

The break-even point in business is when the revenue (the money brought in) is equal to the cost (the money spent to earn the revenue). When a business gets past the break-even point, it shows a profit. To find the break-even point, 

The Problem You decide to go into the sandal-making business. Your startup costs are 1,600, and it costs you 40 per pair of sandals to produce them. You write your total cost function as C = 1,600 + 40x, where x is the number of pairs of sandals that you produce. The price at which you sell the sandals is dependent on the number of pairs you sell, so there isn t a fixed price (you lower the price to be able to sell more). In this case, the amount of revenue you get from selling x pairs of sandals is found with R = lOOx - 0.5x2. What is the break-even point How many pairs of sandals do you have to produce and sell to start making a profit  

Consider some cost and revenue values. Table 17-1 shows the revenue, total cost, and net result for x = 10, 20, 30, and 40 pairs of sandals. 

When 40 pairs of sandals are produced, the revenue and the cost are the same, so x = 40 is the break-even point. 

Making this table is helpful, but it isn t very practical. A better method is to solve the system of equations. The two equations involved are C = 1,600 + 40x and R = lOOx - 0.5x2. Replace the C and the R with y, letting the y represent the amount of money spent in the cost function and the amount of money earned in the revenue function. You want to find out when those two amounts are the same to find the break-even point. Then replace they iny = 1,600 + 40x with lOOx - 0.5X2 from the equation y = lOOx - 0.5X2. Solve for the value of x. 

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