In a circle of radius 10.5cm the minor arc is one-fifth of the major arc. Find the area of the sector corresponding to the major arc.

Explanation

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the relationship between the minor and major arcs.
Given that the minor arc is one-fifth of the major arc, we can denote the length of the major arc as \( X \) cm. Therefore, the length of the minor arc will be \( \frac{X}{5} \) cm.

### Step 2: Calculate the total circumference of the circle.
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Given that the radius \( r = 10.5 \) cm, we can calculate:
\[ C = 2 \times \pi \times 10.5 = 21\pi \text{ cm} \]

### Step 3: Set up the equation for the arcs.
Since the minor arc and the major arc together make up the entire circumference, we can write:
\[ X + \frac{X}{5} = 21\pi \]

### Step 4: Solve for \( X \).
To solve for \( X \), first combine the terms on the left:
\[ \frac{5X + X}{5} = 21\pi \]
\[ \frac{6X}{5} = 21\pi \]
Now, multiply both sides by 5:
\[ 6X = 105\pi \]
Finally, divide by 6:
\[ X = \frac{105\pi}{6} = 17.5\pi \text{ cm} \]

### Step 5: Calculate the area of the sector corresponding to the major arc.
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the central angle in degrees corresponding to the arc length.

### Step 6: Find the central angle \( \theta \) corresponding to the major arc.
The length of an arc is also given by:
\[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \]
Setting this equal to the length of the major arc \( X \):
\[ X = \frac{\theta}{360} \times 2\pi r \]
Substituting \( X = 17.5\pi \) and \( r = 10.5 \):
\[ 17.5\pi = \frac{\theta}{360} \times 2\pi \times 10.5 \]
Dividing both sides by \( \pi \):
\[ 17.5 = \frac{\theta}{360} \times 21 \]
Now, multiply both sides by 360:
\[ 6300 = 21\theta \]
Finally, divide by 21:
\[ \theta = 300 \text{ degrees} \]

### Step 7: Calculate the area of the sector.
Now substitute \( \theta = 300 \) and \( r = 10.5 \) into the area formula:
\[ A = \frac{300}{360} \times \pi \times (10.5)^2 \]
Calculating \( (10.5)^2 = 110.25 \):
\[ A = \frac{5}{6} \times \pi \times 110.25 \]
Calculating:
\[ A = \frac{5 \times 110.25\pi}{6} = \frac{551.25\pi}{6} \approx 288.75 \text{ cm}^2 \]

### Final Answer:
The area of the sector corresponding to the major arc is approximately \( 288.75 \) cm².

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