Business Calculus: Demand/ Marginal Revenue?
Assume that the demand function for tuna in a small coastal town is given by
p = 24,000/q^1.5
(200 ≤ q ≤ 800)
where p is the price (in dollars) per pound of tuna, and q is the number of pounds of tuna that can be sold at the price p in one month.
(a) Calculate the price that the town's fishery should charge for tuna in order to produce a demand of 400 pounds of tuna per month.
$ ___ per lb
(b) Calculate the monthly revenue R (in dollars) as a function of the number of pounds of tuna q.
R(q) =
(c) Calculate the revenue and marginal revenue (derivative of the revenue with respect to q) at a demand level of 400 pounds per month.
revenue $
marginal revenue $ per lb of tuna
Interpret the results.
At a demand level of 400 pounds per month, the revenue is $ and decreasing at a rate of $ per additional pound of tuna.
Comments
(a) Calculate the price that the town's fishery should charge for tuna in order to produce a demand of 400 pounds of tuna per month.
p = 24,000/q^1.5
Substitute q=400 into the equation
p = 24,000 /400^1.5
p = 24,000 /8,000 = $3,000 per lb
b)
Revenue = pq
Revenue = [24,000/q^1.5] q
R(q) = 24,000 / q^0.5
c)
Revenue = 24,000/400^0.5 = 24,000/20 = $1,200
Marginal revenue = R'(q)
R(q) = 24,000 q^(-0.5)
R'(q) = 24,000 (-0.5) q^(-1.5) = -12,000 /q^1.5
R'(400) = -12,000 /400^(1.5) = -12,000/8,000 = -$1.5 per lb of tuna
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