Let 'p' be the price per unit of a certain product, when there is a sale of 'x' units. The total revenue function is :
`P(x)=(100x)/(3x+1)-4x.`
(i) Find the marginal revenue function, rate of change of total revenue function with respect to x.
(ii) When x = 10, find the relative change of revenue R, i.e., Rate of change of R with respect to x and also the percentage rate of change of R at `x=10.`

Answer

To solve the given problem, we will follow the steps outlined in the question.

### Step 1: Find the Marginal Revenue Function

The total revenue function is given by:

\[ P(x) = \frac{100x}{3x + 1} - 4x \]

To find the marginal revenue function, we need to differentiate \(P(x)\) with respect to \(x\).

#### Differentiation

We can differentiate \(P(x)\) using the quotient rule for the first term and the power rule for the second term. The quotient rule states that if you have a function \(h(x) = \frac{f(x)}{g(x)}\), then:

\[ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]

Let \(f(x) = 100x\) and \(g(x) = 3x + 1\).

1. Differentiate \(f(x)\):
\[ f'(x) = 100 \]

2. Differentiate \(g(x)\):
\[ g'(x) = 3 \]

Now apply the quotient rule:

\[ \frac{dP}{dx} = \frac{100(3x + 1) - 100x(3)}{(3x + 1)^2} - 4 \]

Simplifying this:

\[ \frac{dP}{dx} = \frac{100(3x + 1) - 300x}{(3x + 1)^2} - 4 \]
\[ = \frac{300x + 100 - 300x}{(3x + 1)^2} - 4 \]
\[ = \frac{100}{(3x + 1)^2} - 4 \]

Thus, the marginal revenue function \(MR(x)\) is:

\[ MR(x) = \frac{100}{(3x + 1)^2} - 4 \]

### Step 2: Find the Relative Change of Revenue at \(x = 10\)

To find the relative change of revenue, we need to evaluate the marginal revenue at \(x = 10\):

\[ MR(10) = \frac{100}{(3(10) + 1)^2} - 4 \]
\[ = \frac{100}{(30 + 1)^2} - 4 \]
\[ = \frac{100}{31^2} - 4 \]
\[ = \frac{100}{961} - 4 \]
\[ = \frac{100 - 3844}{961} \]
\[ = \frac{-3744}{961} \]

Now, the relative change of revenue \(R\) is given by:

\[ \text{Relative Change} = \frac{MR(10)}{P(10)} \]

First, we need to calculate \(P(10)\):

\[ P(10) = \frac{100(10)}{3(10) + 1} - 4(10) \]
\[ = \frac{1000}{31} - 40 \]
\[ = \frac{1000 - 1240}{31} \]
\[ = \frac{-240}{31} \]

Now substituting into the relative change formula:

\[ \text{Relative Change} = \frac{-3744/961}{-240/31} \]
\[ = \frac{-3744 \times 31}{-240 \times 961} \]
\[ = \frac{116064}{230640} \]
\[ = \frac{116064 \div 116064}{230640 \div 116064} \approx 0.503 \]

### Step 3: Find the Percentage Rate of Change of Revenue

The percentage rate of change of revenue is given by:

\[ \text{Percentage Rate of Change} = \left(\frac{MR(10)}{P(10)}\right) \times 100\% \]
\[ = 0.503 \times 100\% \approx 50.3\% \]

### Final Answers

(i) The marginal revenue function is:

\[ MR(x) = \frac{100}{(3x + 1)^2} - 4 \]

(ii) At \(x = 10\), the relative change of revenue is approximately \(0.503\) and the percentage rate of change of revenue is approximately \(50.3\%\).