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Marginal Revenue

The demand function for a certain boat company's 34 ft Sundancer yacht is:
[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.

(a) Find the revenue function [tex]\( R(x) \)[/tex] and the marginal revenue function [tex]\( R'(x) \)[/tex] for this model of yacht.

[tex]\[ R(x) = x \cdot (900 - 0.01x \ln(x)) \][/tex]
[tex]\[ R'(x) = 900 - 0.02x \ln(x) - 0.01x \][/tex]

(b) Use the result of part (a) to estimate the revenue to be realized from the sale of the 225th 34 ft Sundancer yacht. (Round your answer to the nearest dollar.)

[tex]\[ \$199,792 \][/tex]

Sagot :

GinnyAnswer
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]
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