course Mth 271 015. `query 15*********************************************
.............................................
Given Solution: `a dP/dt=44t + 52 (power function rule on each nonconstant term) When t = 0, 10, 20 and 25 you would have P = 10,000, 12,700, 20,000, 25,000 approx. At these values of t we have dP / dt = 52, 492, 932 and 1152 (these are my mental calculations--check them). dP / dt is the rate of change of the population with respect to time t ** ------------------------------------------------ Self-critique Rating: I took the derivative of the function, evaluated t, then evaluated t in the original function. ********************************************* Question: `q 2.3.48 demand fn p = 50/`sqrt(x), cost .5x+500. Find marginal profit for x=900,1600,2500,3600 Explain how you found the marginal profit, and give your results. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: revenue = p = [50/’sqrt(x)](x) = 50’sqrt(x) cost = .5x + 500 profit = revenue – cost profit = P = (50’sqrt) - .5x – 500 marginal profit = P’ = 25x^(-1/2) - .5 x P 900 .33 1600 .125 2500 0 3600 -.083 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ I feel confident about my answer.
.............................................
Given Solution: `a x represents the number of items sold. If x items are sold at price p = 50 / `sqrt(x), then revenue is price of item * number sold = 50 / `sqrt(x) * x = 50 `sqrt(x). The profit is revenue - cost = 50 `sqrt(x) - .5 x - 500. The marginal profit is the derivative of the profit function, which is (50 `sqrt(x) - .5 x - 500 ) ' = 25 / `sqrt(x) - .5. Evaluating the marginal profit at x = 900, 1600, 2500 and 3600 we get values .33..., .125, 0 and -.0833... . This shows us that the marginal profit, which is the limiting value of the increase in profit per additional item manufactured, is positive until x = 2500. This means that it is to the advantage of the producer to produce new items when x = 900 and when x = 1600, but that the advantage disappears as soon as x reaches 2500. So 2500 is the best selling price. When x = 3600 production of additional items reduces profits. ** ------------------------------------------------ Self-critique Rating: I used logic to find the revenue and the cost. I took the derivative of the sum of these two values and used that value as the marginal profit. From here I plugged in the values of x to get the profit. " xxxx