During the summer months an ice cream van on the seafront in Killiney sells 500

Question

During the summer months an ice cream van on the seafront in Killiney sells 500
cones daily when charging €2 per cone. She finds that increasing the price of a cone will
result in her daily sales falling by 100 cones per I euro increase in price.
(a) Establish the demand equation.
(b) Determine the daily revenue function
(c) Using calculus indicate what price should be charged in order to maximize daily
revenue. Determine also the maximum daily revenue. (16 marks)
(d) The daily cost function for the ice cream vendor has been established to be C(x) :
0.4x + 40 where x is the number of cones and the cost is in €s. Determine the maximum
daily profit and the corresponding price per cone at that profit

Explanation / Answer

a. Demand, p(x) = 500-100x

b. Revenue = Demand* price

Revenue = (500-100x)(2+x) = -100x2+300x + 1000

c. R '(x) = -200x +300

   R '(x) = 0

   -200x + 300 = 0

     x= $ 1.50

d. R(1.50) = -100(1.50) 2+300(1.50) + 1000 = -225 + 450 +1000 = 1225

e . Profit , P(x ) = R(x) - C(x) = -100x2+300x + 1000-0.4x -40 = -100x2+299.6x+ 960

      P '(x) = -200x + 299.6

      P '(x) =0

      -200x + 299.6 =0

        x= 299.6/200 = $1.498

       P(1.498) = -100(1.498)2+299.6(1.498)+ 960 = $1408.008

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