Consider two concentric circles whose circumferences are in a ratio of 4:7. Provided that the area enclosed by the inner circle measures 80 m², calculate the approximate area of the outer circle.

The correct option is 245 m².

Let $r_1$ be the radius of the inner circle and $r_2$ be the radius of the outer circle.

The circumference of a circle with radius $r$ is given by the formula:

$C=2\pi r$

We are given that the ratio of the circumferences of the inner and outer circles is 4:7:

$\frac{2\pi r_1}{2\pi r_2}=\frac{4}{7}$

Simplifying the fraction by cancelling out common terms, we get the ratio of their radii:

$\frac{r_1}{r_2}=\frac{4}{7}$

The area of a circle with radius $r$ is given by the formula:

$A=\pi r^2$

Therefore, the ratio of the areas of the inner circle ($A_1$) and outer circle ($A_2$) is equal to the square of the ratio of their radii:

$\frac{A_1}{A_2}=\frac{\pi r_1^2}{\pi r_2^2}={\left(\frac{r_1}{r_2}\right)}^2$

Substitute the value of $\frac{r_1}{r_2}=\frac{4}{7}$ into the equation:

$\frac{A_1}{A_2}={\left(\frac{4}{7}\right)}^2=\frac{16}{49}$

We are given that the area of the inner circle is $A_1=80$ m². Substituting this value gives:

$\frac{80}{A_2}=\frac{16}{49}$

Now, solve for $A_2$:

$A_2=\frac{80\times 49}{16}$

$A_2=5\times 49=245$ m²

Hence, the area of the outer circle is 245 m².