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
