### Title: Calculating Weekly Profit from Laptop SalesA particular computing company finds that its weekly profit, in dollars, from the production and sale of $ x $ laptop computers is given by the polynomial:\[ P(x) = -0.006x^3 - 0.3x^2 + 500x - 700 \]Currently, the company builds and sells 11 laptops weekly.#### Questions:a) **What is the current weekly profit?**b) **How much profit would be lost if production and sales dropped to 10 laptops weekly?**c) **What is the marginal profit when $ x = 11 $?**d) **Use the answer from part (a) and (c) to estimate the profit resulting from the production and sale of 12 laptops weekly.**---#### Solutions:- **a)** The current weekly profit is $[ \_\_ ] (Round to the nearest cent as needed.)- **b)** The decrease in profit is $[ \_\_ ] (Round to the nearest cent as needed.)- **c)** The marginal profit when $ x = 11 $ is $[ \_\_ ] (Round to the nearest cent as needed.)- **d)** The profit resulting from the production and sale of 12 laptops weekly is approximately $[ \_\_ ] (Round to the nearest cent as needed.)---The polynomial $ P(x) $ helps the company to determine optimal production levels and understand the implications of changing production quantities. The marginal profit is particularly useful for assessing the profitability of producing an additional unit. Remember to round all final answers to the nearest cent for precision.

Profit function is given so we substitute value of x to get profit . For marginal profit derivative of profit function have to be used . Px= -0.006x3-0.3x2+500x -700 a) As company sold 11 laptop currently => x=11 P11= -0.006×113-0.3×112+500×11 -700 = 4755.714 So current weekly profit = 4755.714 b) If production and sales dropped to 10 laptop => x=10 P10= -0.006×103-0.3×102+500×10 -700 = 4264 So dropped in weekly sales = P(11)- P(10)= 4755.714 - 4264 =491.714 Weekly sales dropped by 491.714 C) As marginal profit equal to rate of change in profit Differentiate P(x) with respect to x P'x= -0.006ddxx3-0.3ddxx2+500ddxx -700ddx1 =-0.006×3x2-0.3×2x+500×1-700×0 =-0.018x2-0.6x+500 At x=11 P'11=-0.018× 112-0.6×11+500 = 491.222 So marginal profit when 11 laptop are sold = 491.22d) As marginal profit gives increase in profit when one more laptop are sold P(12) =P(11) +P'(11)= 4755.714+491.22= 5246.934 So profit from production and sales of 12 laptop weekly = 5246.93$