The area of circle and rectangle is 11:6 and the ratio of radius of the circle to length of the rectangle is 1:6. If the breadth of the rectangle is 7 cm, then find the circumference of the circle (in cm).

The correct answer/option is 154.

Let's solve the problem step-by-step.

Step 1: Understand the given ratios and values
Let $r$ be the radius of the circle.
Let $l$ be the length of the rectangle.
Let $b$ be the breadth of the rectangle. We are given that $b=7$ cm.
The ratio of the radius of the circle to the length of the rectangle is given as:
$\frac{r}{l}=\frac{1}{6}$
This means:
$l=6r$

Step 2: Express the area of the circle and the rectangle
The area of a circle with radius $r$ is given by the formula:
$A_{\text{circle}}=\pi r^2$
The area of a rectangle with length $l$ and breadth $b$ is given by:
$A_{\text{rectangle}}=l\times b=\left(6r\right)\times 7=42r$

Step 3: Use the ratio of their areas to find the radius $r$
The ratio of the area of the circle to the area of the rectangle is given as 11:6.
Therefore, we can set up the equation:
$\frac{A_{\text{circle}}}{A_{\text{rectangle}}}=\frac{11}{6}$
Substituting the expressions for the areas into the ratio:
$\frac{\pi r^2}{42r}=\frac{11}{6}$
Since $r\ne 0$, we can simplify by dividing the numerator and denominator on the left side by $r$:
$\frac{\pi r}{42}=\frac{11}{6}$
Now, substitute the value of $\pi \approx \frac{22}{7}$:
$\frac{\left(\frac{22}{7}\right)r}{42}=\frac{11}{6}$
Simplifying the left-hand side:
$\frac{22r}{7\times 42}=\frac{11}{6}$
$\frac{22r}{294}=\frac{11}{6}$
Solve for $r$:
$r=\frac{11}{6}\times \frac{294}{22}$
We can simplify this expression:
$r=\left(\frac{11}{22}\right)\times \left(\frac{294}{6}\right)$
$r=\frac{1}{2}\times 49$
$r=24.5$ cm

Step 4: Find the circumference of the circle
The circumference $C$ of a circle is given by the formula:
$C=2\pi r$
Using $\pi =\frac{22}{7}$ and $r=\frac{49}{2}$:
$C=2\times \frac{22}{7}\times \frac{49}{2}$
The factor of 2 in the numerator and denominator cancels out, and $\frac{49}{7}=7$:
$C=22\times 7$
$C=154$ cm

Thus, the circumference of the circle is 154 cm.