Question

Answer this question. 1.(a) The daily profit, f (x), in hundreds of dollars, for a company that manufactures x radios daily is given by f(x)240x 300 Determine the maximum daily profit of the company. (6 marks) The combined area of a square and a rectangle is 128 cm2. The width of the rectangle is 3 cm more than the length of a side of the square, and the length of the rectangle is 3 cm more than its width. Detemine the dimensions of the square and the rectangle (b) (5 marks) (c)Amanufacturer of tanks makes a tank, with a rectangular base, as shown in the diagram below, where h is the height of the isosceles triangle and x the width of the rectangular base (i) Obtain an expression for the area A of the shaded section (top surface) as a function of the distance x between the parallel sides. (5 marks) (4 marks) Total 20 marks (ii) Hence, find the domain of A (that is, the range of values of x)

Answer 1

(a) f(x) = -x^2 + 40x - 300 For maximum profit of company, d/dx f(x) = 0 d/dx f(x) = -2x + 40 = 0 2x = 40 x = 20 x = 20 is a critical point of f(x) d^2/dx^2 f(x) = -2 At x = 20, d^2/dx^2 f(x) < 0 So, by second derivative test, f(x) has a local maximum at x = 20. f(20) = -(20)^2 + 40 (20) - 300 = -400 + 800 - 300 = 100 For maximum daily profit, company needs to manufacture 20 radios daily and Maximum daily Profit of company = 100 (b) Let length of a side of square = x cm So, width of rectangle = (x + 3) cm and length of rectangle = (x + 3 + 3) = (x + 6) cm