1. The number of units x that consumers are willing to purchase at a given price

Question

1.

The number of units x that consumers are willing to purchase at a given price p is defined as the demand function p=f(x). Suppose the cost of production is given by C(x)=4000- 40x + 0.02x2 and the demand function is p(x)= 50 -x/100. Find the unit price p that produces maximum profit. Find the number of items x for which production cost is minimum. Find the value of x for which average cost C- per item is minimum. (Hint: C- is minimum where its graph has a horizontal tangent line, so the derivative of C- is zero) When average cost C- is minimum, show that average cost and marginal cost are equal. Having produced 1000 items, approximate the additional cost of producing one more. Do the same for 5000 items.

Explanation / Answer

solved 1 part

a)revenue=p*x= 50x -x^2/100

profit P=revenue -cost =(50x -x^2/100)- (4000-40x +0.02x^2)

for max profit P'=0

==>(50- x/50)-(-40+0.04x)=0

==>-0.06x+90=0

==>x=90/0.06=1500

==>p=50- 1500/100

==>p=35------------------------->unit price p that produces maximum profit

  

  

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