The circumference of a certain circle is doubled. By what factor is its area increased?
C=2*pi*r
If you were to double the circumference, since the only variable in the equation is r, that means the radius is double. Since A=pi*r^2, if the radius is double the area is the square of double or quadrupled so it increases by a factor of 4
To determine the factor by which the area of a circle increases when its circumference is doubled, we will need to use some mathematical formulas.
Let's start by recalling the formulas associated with the circumference and area of a circle:
1. Circumference of a circle: C = 2πr, where C represents the circumference and r represents the radius of the circle.
2. Area of a circle: A = πr^2, where A represents the area and r represents the radius of the circle.
Now, let's consider the given scenario, where the circumference of the circle is doubled. Mathematically, this can be expressed as:
New Circumference = 2 * Old Circumference
or,
C' = 2C
We are interested in finding the factor by which the area (A') increases.
To do that, we need to find the new area (A') using the new circumference (C'). We can then compare it to the original area (A) to determine the factor by which it has increased.
Let's substitute the values into the formulas and calculate:
New Circumference: C' = 2C = 2(2πr) = 4πr
Using the formula for the area, we can find the new area (A'):
New Area: A' = πr'^2 = πr^2, since the radius (r) remains the same.
Comparing the new area (A') to the original area (A), we can calculate the factor by which the area is increased:
Factor = (A' - A) / A = (πr^2 - πr^2) / πr^2 = 0.
Hence, we find that when the circumference of a circle is doubled, the area remains the same.