Answer :
Certainly! Let's dive into the solution step-by-step for both parts (a) and (b).
### Part (a): Revenue Function and Marginal Revenue Function
We are given the demand function for the yacht:
[tex]\[ p = 900 - 0.01x \ln(x) \][/tex]
where [tex]\(x\)[/tex] denotes the number of yachts and [tex]\(p\)[/tex] is the price per yacht in hundreds of dollars.
Step 1: Define the Revenue Function [tex]\(R(x)\)[/tex]
Revenue [tex]\(R(x)\)[/tex] is calculated by multiplying the number of yachts [tex]\(x\)[/tex] by the price per yacht [tex]\(p\)[/tex]. Thus, the revenue function is:
[tex]\[ R(x) = x \cdot p \][/tex]
Substituting the given demand function into this formula:
[tex]\[ R(x) = x \cdot (900 - 0.01x \ln(x)) \][/tex]
This simplifies to:
[tex]\[ R(x) = 900x - 0.01x^2 \ln(x) \][/tex]
Step 2: Find the Marginal Revenue Function [tex]\(R'(x)\)[/tex]
The marginal revenue function [tex]\(R'(x)\)[/tex] is the derivative of the revenue function [tex]\(R(x)\)[/tex] with respect to [tex]\(x\)[/tex].
To differentiate [tex]\(R(x) = 900x - 0.01x^2 \ln(x)\)[/tex], use the product rule and chain rule for differentiation:
[tex]\[ R'(x) = \frac{d}{dx}[900x] - \frac{d}{dx}[0.01x^2 \ln(x)] \][/tex]
First term:
[tex]\[ \frac{d}{dx}[900x] = 900 \][/tex]
Second term:
Using the product rule for [tex]\(\frac{d}{dx}[0.01x^2 \ln(x)]\)[/tex]:
[tex]\[ \frac{d}{dx}[0.01x^2 \ln(x)] = 0.01 \left (2x \ln(x) + x \right ) = 0.01 \left (2x \ln(x) + x \right) = 0.02x \ln(x) + 0.01x \][/tex]
Thus, summing these results, we have:
[tex]\[ R'(x) = 900 - (0.02x \ln(x) + 0.01x) \][/tex]
Simplified, this becomes:
[tex]\[ R'(x) = 900 - 0.02x \ln(x) - 0.01x \][/tex]
So, summarizing part (a):
[tex]\[ R(x) = 900x - 0.01x^2 \ln(x) \][/tex]
[tex]\[ R'(x) = 900 - 0.02x \ln(x) - 0.01x \][/tex]
### Part (b): Estimate the Revenue from the Sale of the 225th Yacht
We need to find the revenue [tex]\(R(x)\)[/tex] at [tex]\(x = 225\)[/tex].
Using the revenue function [tex]\(R(x)\)[/tex] derived in part (a):
[tex]\[ R(225) = 900 \cdot 225 - 0.01 \cdot 225^2 \cdot \ln(225) \][/tex]
Given that the calculation has already been done, the rounded revenue at [tex]\(x = 225\)[/tex] is:
[tex]\[ R(225) \approx 199758 \][/tex]
Therefore, the estimated revenue from the sale of the 225th [tex]\(34\)[/tex] ft Sundancer yacht is:
[tex]\[ \boxed{199758} \][/tex]
### Part (a): Revenue Function and Marginal Revenue Function
We are given the demand function for the yacht:
[tex]\[ p = 900 - 0.01x \ln(x) \][/tex]
where [tex]\(x\)[/tex] denotes the number of yachts and [tex]\(p\)[/tex] is the price per yacht in hundreds of dollars.
Step 1: Define the Revenue Function [tex]\(R(x)\)[/tex]
Revenue [tex]\(R(x)\)[/tex] is calculated by multiplying the number of yachts [tex]\(x\)[/tex] by the price per yacht [tex]\(p\)[/tex]. Thus, the revenue function is:
[tex]\[ R(x) = x \cdot p \][/tex]
Substituting the given demand function into this formula:
[tex]\[ R(x) = x \cdot (900 - 0.01x \ln(x)) \][/tex]
This simplifies to:
[tex]\[ R(x) = 900x - 0.01x^2 \ln(x) \][/tex]
Step 2: Find the Marginal Revenue Function [tex]\(R'(x)\)[/tex]
The marginal revenue function [tex]\(R'(x)\)[/tex] is the derivative of the revenue function [tex]\(R(x)\)[/tex] with respect to [tex]\(x\)[/tex].
To differentiate [tex]\(R(x) = 900x - 0.01x^2 \ln(x)\)[/tex], use the product rule and chain rule for differentiation:
[tex]\[ R'(x) = \frac{d}{dx}[900x] - \frac{d}{dx}[0.01x^2 \ln(x)] \][/tex]
First term:
[tex]\[ \frac{d}{dx}[900x] = 900 \][/tex]
Second term:
Using the product rule for [tex]\(\frac{d}{dx}[0.01x^2 \ln(x)]\)[/tex]:
[tex]\[ \frac{d}{dx}[0.01x^2 \ln(x)] = 0.01 \left (2x \ln(x) + x \right ) = 0.01 \left (2x \ln(x) + x \right) = 0.02x \ln(x) + 0.01x \][/tex]
Thus, summing these results, we have:
[tex]\[ R'(x) = 900 - (0.02x \ln(x) + 0.01x) \][/tex]
Simplified, this becomes:
[tex]\[ R'(x) = 900 - 0.02x \ln(x) - 0.01x \][/tex]
So, summarizing part (a):
[tex]\[ R(x) = 900x - 0.01x^2 \ln(x) \][/tex]
[tex]\[ R'(x) = 900 - 0.02x \ln(x) - 0.01x \][/tex]
### Part (b): Estimate the Revenue from the Sale of the 225th Yacht
We need to find the revenue [tex]\(R(x)\)[/tex] at [tex]\(x = 225\)[/tex].
Using the revenue function [tex]\(R(x)\)[/tex] derived in part (a):
[tex]\[ R(225) = 900 \cdot 225 - 0.01 \cdot 225^2 \cdot \ln(225) \][/tex]
Given that the calculation has already been done, the rounded revenue at [tex]\(x = 225\)[/tex] is:
[tex]\[ R(225) \approx 199758 \][/tex]
Therefore, the estimated revenue from the sale of the 225th [tex]\(34\)[/tex] ft Sundancer yacht is:
[tex]\[ \boxed{199758} \][/tex]
