Question

Marginal Revenue
The demand function for a certain boat company's 34 ft Sundancer yacht is
p = 700 − 0.01x ln(x)
where x denotes the number of yachts and p is the price per yacht in hundreds of dollars.
(a) Find the revenue function R(x) and the marginal revenue function R'(x) for this model of yacht.
R(x) = 700x−0.01x2ln(x)
R'(x) = −0.01x−0.02xln(x)+700
(b) Use the result of part (a) to estimate the revenue to be realized from the sale of the 375th 34 ft Sundancer yacht. (Round your answer to the nearest dollar.)

Answer 1

Answer:

Step-by-step explanation:

a) Revenue function us derives by taking the product of the number of yachts and the demand function p given. Mathematically,

R(x) = xp(x)

Given p(x) = 700 − 0.01x ln(x)

R(x) = x{700 − 0.01x ln(x)}

R(x) = 700x - x(0.01x ln(x))

R(x) = 700x - 0.01x²lnx

Hence the revenue function R(x) is expressed as R(x) = 700x - 0.01x²lnx

Marginal revenue function is derived by finding the derivative of the revenue function R(x). On differentiating;

d{R(x)}/dx = 700 - 0.01{x²(1/x)+2xlnx} (note that product rule was used to differentiate the function in parenthesis).

d{R(x)}/dx = 700 - 0.01{x+2xlnx}

Open the parenthesis

d{R(x)}/dx = 700 - 0.01x-0.02xlnx

Hence the marginal revenue function R'(x) is expressed as 700 - 0.01x-0.02xlnx.

b) In order to estimate the revenue to be realized from the sale of the 375th 34 ft Sundancer yacht, we will simply substitute the variable x = 375 into the revenue function R(x)

Given R(x) = 700x - 0.01x²lnx

R(375) = 700(375) - 0.01(375)²ln(375)

R(375) = 262500-1406.25ln(375)

R(375) = 262500-8334.74

R(375) = 254,165.26

Hence the revenue realized from the sale is approximately $254,165

For the Marginal revenue Function,

R'(x) = 700−0.01x−0.02xln(x)

R'(375) = 700−0.01(375)−0.02(375)ln(375)

R'(375) = 700−3.75−0.02(375)ln(375)

R'(375) = 700-3.75-44.45

R'(375) = 651.8

The marginal revenue is approximately $652