A manufacturer has been selling 1000 television sets a week at $500 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 100 per week. Round your answers to the nearest dollar.(a) Find the linear demand function (price as a function of units sold).(b) How large a rebate should the company offer the buyer in order to maximize its revenue?(c) If the company experiences a cost of C(x) = 76,000 + 110x, how should the manufacturer set the size of the rebate in order to maximize its profit?

Answer:a) The linear function is P(x) = 600 - x/10 b) $200c) $145Step-by-step explanation:The manufacturer sold a total number of 1000 television in a week The selling price of each television is $500The sale increases by 100 per week if a rebate of $10 dollars is offered to the buyer. Therefore, we have a price decrease of 1/100 * 10 = 1/10 for each unitLet x be the number sold per week. x-1000 gives the increase in salesLet P(x) be the priceP(x) = 500 - 1/10(x - 1000) = 500 - x/10 + 1000/10 = 500 - x/10 + 100Collect like terms P(x) = 500 +100 - x/10 = 600 - x/10The linear demand function with price as a function of units sold is given asP(x) = 600 - x/10b) Let R(x) be the revenue Revenue = number sold * Price R(x) = x* P(x)R(x) = x(600 -x/10) = 600x - x^2/10Differentiate R(x) with respect to yR'(x) = 600 - 2x/10 = 600 - x/5The revenue is maximum when R(x) = 0600 - x/5 = 0600 = x/5x = 600*5x = 3000Remember that P(x) = 600 - x/10P(3000) = 600 - 3000/10 = 600 - 300 = 300The rebate to maximize the revenue will be 500 - 300 = $200c) C(x) = 76000 + 110x C(x) = CostP(x) = R(x) - C(x)P(x) = (600x - x^2/10) - (76000 + 110x) = 600x - x^2/10 - 76000 - 110xCollect like terms P(x) = 600x - 110x - x^2/10 - 76000 = 490x - x^2/10 - 76000Differentiate P(x) with respect to x P'(x) = 490 - 2x/10 = 490 - x/5P(x) is maximized when it is equal to 0490 - x/5 = 0490 = x/5x = 490*5x = 2450P(x) = 600 - x/10P(2450) = 600 - 2450/10 = 600 -245 = 355The rebate to maximize the revenue will be 500 - 355 = $145