The area of a circle is given by the formula A = pi*r^2 and the circumference of the circle is given by C = 2*pi*r. Find the ratio of the area to the circumference. What happens to this ratio if the radius is quadrupled?
Thank you!
A/C = pi(r^2)/(2pi r)
= r/2
if radius is quadrulpled
A = pi(4r)^2 = 16pi(r^2)
C = 2pi(4r) = 8pi r
ratio A/C = 16pi r^2/(8pi r) = 2r
To find the ratio of the area to the circumference, we can divide the area formula by the circumference formula:
Ratio = A / C = (pi*r^2) / (2*pi*r)
Simplifying the equation, we can cancel out the pi:
Ratio = r^2 / (2*r)
Now, let's examine what happens to this ratio if the radius is quadrupled.
If the radius is quadrupled, the new radius will be 4 times the original radius. Let's call the new radius R.
Substituting R = 4r into the ratio formula:
New Ratio = (R^2) / (2*R) = (16r^2) / (8r) = 2r
The new ratio simplifies to 2r.
Therefore, if the radius is quadrupled, the ratio of the area to the circumference becomes 2r.
To find the ratio of the area to the circumference of a circle, we can substitute the formulas for area (A) and circumference (C) into the following equation:
Ratio = A / C
Using the given formulas, we have:
Ratio = (pi * r^2) / (2 * pi * r)
The value of pi (π) cancels out:
Ratio = r^2 / (2r)
Simplifying the expression further:
Ratio = r / 2
Now, let's consider what happens to this ratio if the radius is quadrupled. If the original radius is 'r', then the new radius after quadrupling would be '4r'.
Substituting '4r' for 'r' in the ratio equation:
New Ratio = (4r) / 2
New Ratio = 2r
Therefore, if the radius is quadrupled, the new ratio of the area to the circumference becomes 2 times the original ratio.