The length of the perimeter of a sector of a circle is 20 cm. Give an expression for the area of the sector in terms of r(the radius of the circle) and hence find the maximum area of the sector.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Understanding the Perimeter of the Sector
The perimeter of a sector of a circle is given by the formula:
\[ \text{Perimeter} = 2r + r\theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians. We are given that the perimeter is 20 cm:
\[ 2r + r\theta = 20 \]

### Step 2: Rearranging the Equation
We can rearrange this equation to express \( r \) in terms of \( \theta \):
\[ r(2 + \theta) = 20 \]
\[ r = \frac{20}{2 + \theta} \tag{1} \]

### Step 3: Finding the Area of the Sector
The area \( A \) of the sector can be expressed as:
\[ A = \frac{1}{2} r^2 \theta \]
Substituting the expression for \( r \) from equation (1):
\[ A = \frac{1}{2} \left(\frac{20}{2 + \theta}\right)^2 \theta \]
\[ A = \frac{1}{2} \cdot \frac{400}{(2 + \theta)^2} \cdot \theta \]
\[ A = \frac{200\theta}{(2 + \theta)^2} \]

### Step 4: Maximizing the Area
To find the maximum area, we need to differentiate \( A \) with respect to \( \theta \) and set the derivative equal to zero:
\[ \frac{dA}{d\theta} = 0 \]
Using the quotient rule:
\[ \frac{dA}{d\theta} = \frac{(2 + \theta)^2 \cdot 200 - 200\theta \cdot 2(2 + \theta)}{(2 + \theta)^4} \]
Setting the numerator equal to zero:
\[ 200(2 + \theta)^2 - 400\theta(2 + \theta) = 0 \]
\[ 200(2 + \theta)(2 + \theta - 2\theta) = 0 \]
\[ 200(2 + \theta)(2 - \theta) = 0 \]
This gives us \( \theta = 2 \) (since \( 2 + \theta \neq 0 \)).

### Step 5: Confirming Maximum Area
To confirm that this is a maximum, we can check the second derivative or simply evaluate the first derivative around \( \theta = 2 \). However, we can directly substitute \( \theta = 2 \) back into the area formula:
\[ r = \frac{20}{2 + 2} = \frac{20}{4} = 5 \]
Now substituting \( \theta = 2 \) into the area formula:
\[ A = \frac{200 \cdot 2}{(2 + 2)^2} = \frac{400}{16} = 25 \text{ cm}^2 \]

### Final Result
The maximum area of the sector is:
\[ \boxed{25 \text{ cm}^2} \]