A candy vendor analyzes his sales records and finds that if he sells x candy bars in one day, his profit(in dollars) is given byP(x) = − 0.001x2 + 3x − 1800 (a.) Explain the significance of the number 1800 to the vendor. (b.) What is the maximum profit he can make in one day, and how many candy bars must he sell to achieve it? I got 1800 as the amount he starts with, and can't go over. maximum profit as 4950 and if I got that right I am getting stuck on how to find how many candy bars. Thanks
a) 1800 is the cost to run the business for a day. To clarify, when you plug in x = 0 for 0 candy bars sold, you are left with -1,800, which is the cost of doing business for one day. b) Maximum profit is found by taking the derivative of the profit equation and setting it equal to 0. P'(x) = -0.002x + 3 With P'(x) = 0, we get: -0.002x + 3 = 0 Using our equation solver, we get: x = 1,500 To get maximum profit, we plug in x = 1,500 to our original profit equation P(1,500) = − 0.001(1,500)^2 + 3(1,500) − 1800 P(1,500) = -2,250 + 4,500 - 1,800 P(1,500) = $450
