Please find attached the mathematics questions.

Question 1 (a) The cost of producing 5-gallon water bottles is given by C(q) = 0.005q2 + 2g + 1000. If 2000 5-gallon water bottles are produced, find the (i) total cost (ii) average cost (iii) marginal cost (iv) marginal average cost szs if xsz (b) Suppose that f(x) = {BX + 2 if x > 2 Find the value of B so that f(x) is continuous everywhere along the real line. (c) The demand for watches is given by p = 7000 2q dollars, and the supply ofwatches is given byp = 0.01q2 + 2q + 1000 dollars, where q is the number of watches demanded and supplied when the price per watch is p dollars. Based on the information provided, calculate the equilibrium quantity and the equilibrium price for watches. (d) Show that X2 4- is a factor of x4 5x3 + 2X2 + 20x 24. Page 2 Question 2 (a) For matrixA = (4 1 2 3) , ifg(X) = X2 + 4x + 2, what is the answer to g(A)? (b) Using algebra, nd the solution to the system of inequalities below y S 5 3x y + 19 2 x2 8x (c) What is peculiar about the slope of a function before and after a point of inection as compared with the slope before or after a maximum or minimum point? (d) Suppose that the satifaction received from a number of rides taken on a roller coaster 2x+1 3x+4 0.5 is given by the function SCX) = 200 ( ) , where S is the satisfaction received and X is the number of rides taken. (i) Find the rate at which the satisfaction changes with respect to the number of rides taken. Question 3. (a) A project management consultant estimated that if a particular project was completed, it years after completion, N thousand persons would benefit directly from the project, where t3 N(t) =4.5t2 +16t, 0 s t s 10 For what value of t, will the largest number of people receive direct benefits? (b) A clothing manufacturer makes trousers, skirts and blouses. Each trouser requires 20 minutes ofcutting time, 60 minutes of sewing time and 5 minutes ofpackaging time. Each skirt requires 15 minutes of cutting time, 30 minutes of sewing time and 12 minutes of packaging time. Each blouse requires 10 minutes of cutting time, 24 minutes of sewing time and 6 minutes of packaging time. The amount of time available for cutting, sewing and packaging is 115 hours, 280 hours and 65 hours respectively. Using either the Inverse Method or the Cram er's Rule, determine how many of each type of clothing should be made to use all available labour hours? Page 3 Question 4- (a) Find the following: _ L. x1/3 + 9x1/5 __ L_ (2x 1)1/2 3 1' Slim '1' $ng (b) The total revenue curve of a firm is R(q) = 4-0q 12g2 and its 400 1 average cost ACq) = qu 12.85q + 20 + , where q is the firm's output. q i. Derive an expression C(q) for the firm's total cost function. ii. Derive an expression l'l(q) for the firm's prot function. iii. 15 the rate of change of profit increasing or decreasing when the ouput level of the firm is 10 units? iv. Determine the level of output for which the firm's prot is maximized. v. What is the firms's maximum profit? END OF A SSIGNMENT