The difference between the circumference of circular field X and circular field Y is 176 m. If the radius of circular field Y is 5 times the radius of circular field X, then find the area of circular field Y (in sq. m)?

The correct answer is 3850.

Step 1: Understand the given formulas and relations
Let $r_{\text{X}}$ be the radius of circular field X, and $r_{\text{Y}}$ be the radius of circular field Y.
The circumference of a circle with radius $r$ is given by:

$\text{Circumference}=2\pi r$

Given that the radius of circular field Y is 5 times the radius of circular field X:

$r_{\text{Y}}=5r_{\text{X}}$

Step 2: Use the difference in circumferences to find the radius
The difference between the circumference of field Y and field X is 176 m:

$2\pi r_{\text{Y}}-2\pi r_{\text{X}}=176$

Substitute $r_{\text{Y}}=5r_{\text{X}}$ into the equation:

$2\pi \left(5r_{\text{X}}\right)-2\pi r_{\text{X}}=176$

$10\pi r_{\text{X}}-2\pi r_{\text{X}}=176$

$8\pi r_{\text{X}}=176$

Substitute $\pi =\frac{22}{7}$:

$8\times \frac{22}{7}\times r_{\text{X}}=176$

$\frac{176}{7}\times r_{\text{X}}=176$

$r_{\text{X}}=7\text{m}$

Step 3: Calculate the radius of circular field Y
Using $r_{\text{Y}}=5r_{\text{X}}$:

$r_{\text{Y}}=5\times 7=35\text{m}$

Step 4: Calculate the area of circular field Y
The area of a circle is given by $\text{Area}=\pi r^2$:

$\text{Area of Y}=\pi {\left(r_{\text{Y}}\right)}^2$

$\text{Area of Y}=\frac{22}{7}\times {35}^2$

$\text{Area of Y}=\frac{22}{7}\times 35\times 35$

$\text{Area of Y}=22\times 5\times 35=3850\text{sq. m}$

Thus, the area of circular field Y is 3850 sq. m.