The arcs of two circles whose radii are in the ratio of `4 : 3` subtend an angle of `48^(@)` each at their centres. Compare the areas of the two sectors

Answer

To compare the areas of the two sectors formed by the arcs of two circles with given radii and angles, we can follow these steps:

### Step 1: Define the Radii
Let the radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \). According to the problem, the ratio of the radii is given as:
\[ \frac{r_1}{r_2} = \frac{4}{3} \]

### Step 2: Express the Radii
From the ratio, we can express the radii in terms of a common variable. Let:
\[ r_1 = 4k \quad \text{and} \quad r_2 = 3k \]
where \( k \) is a positive constant.

### Step 3: Use the Formula for Area of a Sector
The area \( A \) of a sector of a circle is given by the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.

### Step 4: Calculate the Area of the First Sector
For the first circle with radius \( r_1 \) and angle \( 48^\circ \):
\[ A_1 = \frac{48}{360} \times \pi (r_1)^2 = \frac{48}{360} \times \pi (4k)^2 \]
Calculating \( (4k)^2 \):
\[ A_1 = \frac{48}{360} \times \pi \times 16k^2 \]
Simplifying:
\[ A_1 = \frac{48 \times 16}{360} \pi k^2 = \frac{768}{360} \pi k^2 = \frac{64}{30} \pi k^2 = \frac{32}{15} \pi k^2 \]

### Step 5: Calculate the Area of the Second Sector
For the second circle with radius \( r_2 \) and angle \( 48^\circ \):
\[ A_2 = \frac{48}{360} \times \pi (r_2)^2 = \frac{48}{360} \times \pi (3k)^2 \]
Calculating \( (3k)^2 \):
\[ A_2 = \frac{48}{360} \times \pi \times 9k^2 \]
Simplifying:
\[ A_2 = \frac{48 \times 9}{360} \pi k^2 = \frac{432}{360} \pi k^2 = \frac{12}{10} \pi k^2 = \frac{6}{5} \pi k^2 \]

### Step 6: Compare the Areas of the Two Sectors
Now, we can find the ratio of the areas \( A_1 \) and \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{\frac{32}{15} \pi k^2}{\frac{6}{5} \pi k^2} \]
The \( \pi k^2 \) cancels out:
\[ \frac{A_1}{A_2} = \frac{32/15}{6/5} = \frac{32}{15} \times \frac{5}{6} = \frac{32 \times 5}{15 \times 6} = \frac{160}{90} = \frac{16}{9} \]

### Conclusion
The ratio of the areas of the two sectors is:
\[ A_1 : A_2 = 16 : 9 \]