Question

The circumference of a circle is directly proportional to the length of its radius, while the area it encloses is directly proportional to the square of the length of its radius. If the area of a circle is multiplied by 49, then its circumference is multiplied by ______.

Answer 1

Answer:

If the area of a circle is multiplied by 49, then its circumference is multiplied by 7.

Step-by-step explanation:

Let r represent radius of circle.

We have been given that the circumference of a circle is directly proportional to the length of its radius.

$C=\pi r$, where,

C = Circumference of circle,

r = Radius.

We are also told that the area it encloses is directly proportional to the square of the length of its radius.

$A=\pi r^2$, where A represents area of circle and r represents radius.

$\pi$ is constant of proportionality in both formulas.

We are also told that the area it encloses is directly proportional to the square of the length of its radius. We can represent this information as:

$A=\pi*49$

The radius of circle would be square root of 49.

$r^2=49\\\\r=7$

Since radius of circle is 7, therefore, its circumference is multiplied by 7.

Answer 2

If the area of a circle is multiplied by 49, then its circumference is multiplied by 7.

What is the circumference of the circle?

The circumference of the circle is the length of its boundary, also known as the perimeter of the circle.

As it is given that the circumference of a circle is directly proportional to the length of its radius, therefore, the relationship can be written as,

$\rm \text{Circumference of Circle} \propto \text{Length of its radius}\\\\Circumference \propto r$

Now, further, it is given that the area it encloses is directly proportional to the square of the length of its radius.

$\rm \text{Area of Circle} \propto \text{(Length of its radius)}^2\\\\Area\propto r^2\\\\r \propto \sqrt{Area}$

Now, as we can see that the length of the radius is the square root of the area, therefore, if the area is multiplied by 49, then the circumference is multiplied by 7.

$r \propto \sqrt{Area \times 49}\\\\r \propto 7\sqrt{Area}$

And, the length of the circle is directly proportional to the circumference of the circle, therefore, the circumference of the circle can be written as,

$\rm Circumference \propto r\\\\7(Circumference) \propto 7r$

Hence, If the area of a circle is multiplied by 49, then its circumference is multiplied by 7.

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