Question

If the radii of two wheels are in the ratio $3:7,$ then the ratio of their areas is:

• A. $9:49$
• B. $49:9$
• C. $6:14$
• D. $3:49$

Answer 1

The correct option is A $9:49$

Given, the ratio of the radii of two wheels is $3:7.$

Let the radius of the first wheel be $3x,$ then the radius of the second wheel will be $7x.$
We know that, $\text{Area of a circle =}\pi \times \text{(radius)}{}^2$

Step 1: Area of the first wheel $=\pi \times {\text{(radius of the first wheel)}}^2$
$=\pi \times (3x{)}^2$
$=\pi \times 3^2\times x^2$
$=\pi \times 9\times x^2$
$=9\pi x^2$

Step 2: Area of the second wheel $=\pi \times {\text{(radius of the second wheel)}}^2$
$=\pi \times (7x{)}^2$
$=\pi \times 7^2\times x^2$
$=\pi \times 49\times x^2$
$=49\pi x^2$

Step 3: Ratio of the areas
$=\text{Area of the first wheel : Area of the second wheel}$
$=9\pi x^2:49\pi x^2$
Divide both sides by $\pi x^2$
$=\frac{9\pi x^2}{\pi x^2}:\frac{49\pi x^2}{\pi x^2}$
$=9:49$

Hence, option (a.) is the correct choice.