The amount of pollution content added in a ir in a city due to x-diesel vehicles is given by `p(x)= 0.005x^(3) + 0.02x^(2) + 30x`. Find the marginal increase in pollution content when 3 diesel vehicles are added.

Answer

To find the marginal increase in pollution content when 3 diesel vehicles are added, we need to follow these steps:

### Step 1: Define the function for pollution content
The amount of pollution content added in the air due to \( x \) diesel vehicles is given by the function:
\[ p(x) = 0.005x^3 + 0.02x^2 + 30x \]

### Step 2: Differentiate the function
To find the marginal increase in pollution content, we need to differentiate the function \( p(x) \) with respect to \( x \):
\[ p'(x) = \frac{d}{dx}(0.005x^3 + 0.02x^2 + 30x) \]

Using the power rule of differentiation, where \( \frac{d}{dx}(x^n) = nx^{n-1} \), we differentiate each term:
- The derivative of \( 0.005x^3 \) is \( 3 \cdot 0.005x^{3-1} = 0.015x^2 \)
- The derivative of \( 0.02x^2 \) is \( 2 \cdot 0.02x^{2-1} = 0.04x \)
- The derivative of \( 30x \) is \( 30 \)

Thus, we have:
\[ p'(x) = 0.015x^2 + 0.04x + 30 \]

### Step 3: Evaluate the derivative at \( x = 3 \)
Now, we need to find the marginal increase in pollution content when 3 diesel vehicles are added, which means we evaluate \( p'(3) \):
\[ p'(3) = 0.015(3^2) + 0.04(3) + 30 \]

Calculating each term:
- \( 3^2 = 9 \), so \( 0.015 \cdot 9 = 0.135 \)
- \( 0.04 \cdot 3 = 0.12 \)
- The constant term is \( 30 \)

Now, summing these values:
\[ p'(3) = 0.135 + 0.12 + 30 = 30.255 \]

### Final Answer
Thus, the marginal increase in pollution content when 3 diesel vehicles are added is:
\[ \boxed{30.255} \]

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