If the marginal revenue of a commodity is given by MR `=20e^(-x/10)(1-x/10)` , find the demand function .

Answer

To find the demand function from the given marginal revenue function \( MR = 20 e^{-x/10} \left(1 - \frac{x}{10}\right) \), we will follow these steps:

### Step 1: Write the Marginal Revenue as a Differential Equation
The marginal revenue \( MR \) can be expressed as:
\[ MR = \frac{dR}{dx} = 20 e^{-x/10} \left(1 - \frac{x}{10}\right) \]

### Step 2: Integrate to Find the Revenue Function
To find the revenue function \( R \), we need to integrate the marginal revenue:
\[ dR = 20 e^{-x/10} \left(1 - \frac{x}{10}\right) dx \]
Thus, we can write:
\[ R = \int 20 e^{-x/10} \left(1 - \frac{x}{10}\right) dx \]

### Step 3: Simplify the Integral
We can simplify the integral by substituting:
Let \( t = -\frac{x}{10} \) which implies \( dx = -10 dt \). Therefore, we can rewrite the integral:
\[ R = \int 20 e^{t} \left(1 + t\right)(-10 dt) \]
This simplifies to:
\[ R = -200 \int e^{t} (1 + t) dt \]

### Step 4: Solve the Integral
Now we can solve the integral:
\[ \int e^{t} (1 + t) dt = e^{t}(1 + t) - \int e^{t} dt = e^{t}(1 + t) - e^{t} + C \]
Thus, we have:
\[ R = -200 \left( e^{t}(t) \right) + C \]

### Step 5: Substitute Back for \( t \)
Substituting back \( t = -\frac{x}{10} \):
\[ R = -200 \left( e^{-x/10} \left(-\frac{x}{10}\right)\right) + C \]
This simplifies to:
\[ R = 20 x e^{-x/10} + C \]

### Step 6: Determine the Constant \( C \)
To find the constant \( C \), we use the fact that when \( x = 0 \), \( R = 0 \):
\[ 0 = 20(0)e^{0} + C \implies C = 0 \]
Thus, the revenue function is:
\[ R = 20 x e^{-x/10} \]

### Step 7: Find the Demand Function
The demand function \( D \) is given by:
\[ D = \frac{R}{x} = \frac{20 x e^{-x/10}}{x} = 20 e^{-x/10} \]

### Final Answer
Therefore, the demand function is:
\[ D = 20 e^{-x/10} \]