Topic 11: Linear Programming You manage a small branch of your bank. You have de

Question

Topic 11: Linear Programming You manage a small branch of your bank. You have developed a simplified total cost function given by C =20+4X and a revenue function given by R =22X-4X^2 , where X is the level of activity (in hundreds). C and R are both in units of $’000. A. Derive the expression for the total profit, P. B. Calculate the level of activity that maximizes profit and the amount of profit at that level of activity. C. What is your branch’s profit situation when the activity level is 350 units?

Explanation / Answer

Revenue function is given by R(x) = 22x -4x^2

Cost function C(x) = 20 + 4x

(A) Total profit P(x) = R(x) - C(x) = 22x - 4x^2 - 20 - 4x............................................Ans

(B) For maximum profit P ' (x) = 0 ==> 22 - 8x - 4= 0 and P '' (x) <0

x = 18/8 (in hundred)

x = (18/8)*100= 225........................................................Ans

P " (x) = -4 <0 So x =225 indeed is lcal maxima.

(C) X = 350 But in the expression it is in hundreds so x =3.5

P(3.5) = 22 * 3.5 - 4* 3.5*3.5 - 20 = 8 ( in thousands )

So profit at 350 unit = 8000 $...........................................................................................Ans

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