Question

If the radius of an expanding circle is increasing half as quickly as the area is increasing, the circumference at that moment is:

Options:
a) Increasing at the same rate as the radius.
b) Increasing at twice the rate of the radius.
c) Increasing at the same rate as the area.
d) Increasing at four times the rate of the radius.

Answer 1

The circumference at the moment when the radius of an expanding circle is increasing half as quickly as the area is increasing is increasing at the same rate as the radius.

Step-by-step explanation:

To find the relationship between the radius and the circumference of an expanding circle, we need to understand how the area of a circle changes with respect to the radius. It is given that the radius of the circle is increasing half as quickly as the area is increasing. Let's assume the rate at which the radius is increasing is 'r' units per minute. Then, the rate at which the area is increasing would be '2r' square units per minute.

The formula for the area of a circle is A = πr^2. So, if the radius is increasing at 'r' units per minute, the rate of increase in area would be

1. Initial rate of increase in area = πr^2
2. Rate of increase in area when radius is increasing at 'r' units per minute = π(r+r)^2 = π(2r)^2 = 4πr^2

Now, we know that the circumference of a circle is given by the formula C = 2πr. To find the rate of increase in the circumference, we substitute the rate of increase in the radius:

1. Initial rate of increase in circumference = 2πr
2. Rate of increase in circumference when radius is increasing at 'r' units per minute = 2π(2r) = 4πr

Therefore, the circumference at that moment is increasing at the same rate as the radius. So, option a) is correct.