Question

A cruise line estimates that it can sell 3,400 tours to Alaska at a price of $900 each, but it will lose 100 sales for each $50 increase in its price. Let p be the price and q the number of tours sold.

(a) Write q as a linear function of p.

(b) Write the revenue R as a function of p and find the marginal revenue with respect to p (i.e., the rate of change of R with respect to p).

(c) Suppose it costs $400 per passenger to operate the tour. Write the cost as a function of the price p and find the marginal cost with respect to p.

(d) Find the marginal profit with respect to p.

(e) The company priced its tour at $1,050 last year. If it increases the price this year, will it increase or decrease its profit? Explain your answer.

(f) Answer the same question and explain your answer if last year’s price was $1,550.

Answer 1

(A)

Linear function equation:

q = m*p + c

at 3400 quantity price was $900, so 3400 = m*900 + c ...(i)

Now, marginal revenue, q*p = derivative of (m*p*p + c*p) wrt price ,p

pq' = 2m*p +c

as according to question will loose $100 sales for $50 price, so, q1 - q2 = 100 = 50m ; m = 2 ....(ii)

using i & ii to find the value of m & c we get, m = 2 ; c =1600

q = 2p + 1600

(B)

pq = 2*p^2 +p*1600

marginal revenue: pq' = 4*p +1600

(C)

Cost as a function of tourist: C = 400*q

substituting the q from the above mentioned equation, we get:

C = 400*(2p + 1600)

C= 800p + 640,000

C' = 800

(D)

Profit = Revenue - Cost = p*q - C = 2*p^2 +p*1600 - 800p - 640,000 = 2*p^2 + 800*p - 640,000

(E)

Price of per tour is = $1050 = p

For Quadratic Equation curve is something is like following:

So the minimum value can achieved at the p= -b/2a = -800/2*2 = -200

But price can't be negative. Also any $ price right side or above -$200(if possible negative price) then profit will increase.

Explaination: courve is upward sloping from its bottom position which is at -$200 = p which is not possible so it is evident that half of the curve is at the negative side other half is at positive side so in increase in the positive side will mean an increase in profit.

Price of per tour is = $1050 = p profit is = $2,405,000

new price p = $1100 profit is = $2,660,000

Therefore an increase in profit as price is increased

(F)

Same answer as last one. Profit is positively proportional to Price.

Increase in Price lead to Increase in Profit & vice cersa.