The semi-vertical angle of a cone is `45^(@)` . If the height of the cone is 20.025, then its approximate laternal surface area, is

Options

A. `401sqrt(2) pi`
B. `400sqrt(2) pi`
C. `399sqrt(2) pi`
D. none of these

Explanation

To find the approximate lateral surface area of a cone with a semi-vertical angle of \(45^\circ\) and a height of \(20.025\), we can follow these steps:

### Step 1: Understand the Geometry of the Cone
Given that the semi-vertical angle is \(45^\circ\), we can deduce that the radius \(R\) of the cone is equal to its height \(H\). This is because in a right triangle formed by the radius, height, and slant height, if the angle is \(45^\circ\), then:

\[ H = R \]

### Step 2: Use the Pythagorean Theorem
The slant height \(L\) can be calculated using the Pythagorean theorem:

\[ L^2 = H^2 + R^2 \]

Since \(H = R\), we can substitute \(R\) with \(H\):

\[ L^2 = H^2 + H^2 = 2H^2 \]

Thus,

\[ L = H\sqrt{2} \]

### Step 3: Define the Lateral Surface Area Formula
The formula for the lateral surface area \(S\) of a cone is given by:

\[ S = \pi R L \]

Substituting \(R\) and \(L\) in terms of \(H\):

\[ S = \pi H (H\sqrt{2}) = \pi H^2 \sqrt{2} \]

### Step 4: Substitute the Height
Given that the height \(H\) is \(20.025\), we can express \(S\) as:

\[ S = \pi (20.025)^2 \sqrt{2} \]

### Step 5: Calculate \(S\)
First, calculate \(20.025^2\):

\[ 20.025^2 = 400.100625 \]

Now substitute this value into the equation for \(S\):

\[ S = \pi \cdot 400.100625 \cdot \sqrt{2} \]

### Step 6: Approximate the Change in Height
To find the approximate change in lateral surface area due to a small change in height, we denote the change in height as \(\Delta H = 0.025\). We need to find the derivative of \(S\) with respect to \(H\):

\[ \frac{dS}{dH} = 2\pi H \sqrt{2} \]

Now evaluate this at \(H = 20\):

\[ \frac{dS}{dH} \bigg|_{H=20} = 2\pi (20) \sqrt{2} = 40\pi \sqrt{2} \]

### Step 7: Calculate the Change in Surface Area
The change in surface area \(\Delta S\) can be calculated as:

\[ \Delta S = \frac{dS}{dH} \cdot \Delta H = 40\pi \sqrt{2} \cdot 0.025 = \pi \sqrt{2} \cdot 1 \]

### Step 8: Final Approximate Lateral Surface Area
Now, we can find the approximate lateral surface area:

\[ S_{\text{approx}} = S + \Delta S \]

Substituting the values we have:

\[ S_{\text{approx}} = \pi \cdot 400 \sqrt{2} + \pi \sqrt{2} = \pi \sqrt{2} (400 + 1) = 401\pi \sqrt{2} \]

### Conclusion
Thus, the approximate lateral surface area of the cone is:

\[ \boxed{401\pi\sqrt{2}} \]