Find the difference of the area of a sector of angle `90^@` and its coresponding major sector of a circle of radius 9.8 cm

Answer

To find the difference between the area of a sector of angle \(90^\circ\) and its corresponding major sector in a circle of radius \(9.8\) cm, we can follow these steps:

### Step 1: Calculate the area of the smaller sector
The formula for the area of a sector is given by:

\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]

where \(\theta\) is the angle of the sector and \(r\) is the radius. For the smaller sector with \(\theta = 90^\circ\):

\[ \text{Area of smaller sector} = \frac{90}{360} \times \pi \times (9.8)^2 \]

### Step 2: Calculate the area of the major sector
The angle of the major sector is the remainder of the circle, which is:

\[ 360^\circ - 90^\circ = 270^\circ \]

Using the same formula for the area of a sector, we have:

\[ \text{Area of major sector} = \frac{270}{360} \times \pi \times (9.8)^2 \]

### Step 3: Find the difference between the areas
Now, we need to find the difference between the area of the major sector and the area of the smaller sector:

\[ \text{Difference} = \text{Area of major sector} - \text{Area of smaller sector} \]

Substituting the areas we calculated:

\[ \text{Difference} = \left(\frac{270}{360} \times \pi \times (9.8)^2\right) - \left(\frac{90}{360} \times \pi \times (9.8)^2\right) \]

### Step 4: Factor out common terms
We can factor out the common terms:

\[ \text{Difference} = \pi \times (9.8)^2 \left(\frac{270 - 90}{360}\right) \]

### Step 5: Simplify the expression
Now, simplifying the expression:

\[ \text{Difference} = \pi \times (9.8)^2 \left(\frac{180}{360}\right) = \pi \times (9.8)^2 \times \frac{1}{2} \]

### Step 6: Calculate the numerical value
Now we can substitute \(\pi \approx \frac{22}{7}\):

\[ \text{Difference} \approx \frac{22}{7} \times (9.8)^2 \times \frac{1}{2} \]

Calculating \( (9.8)^2 \):

\[ (9.8)^2 = 96.04 \]

Now substituting this value:

\[ \text{Difference} \approx \frac{22}{7} \times 96.04 \times \frac{1}{2} \]

Calculating further:

\[ \text{Difference} \approx \frac{22 \times 96.04}{14} \approx \frac{2112.88}{14} \approx 150.92 \text{ cm}^2 \]

### Final Answer
The difference of the area of the sector of angle \(90^\circ\) and its corresponding major sector is approximately \(150.92 \text{ cm}^2\).

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