Answer:
1. the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm
2. the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm
Explanation:
We can use the formulas for the area and circumference of a circle to solve these problems:
a. To find how fast the radius is changing when the radius is 3 cm, we can use the formula for the area of a circle:
A = πr^2
Taking the derivative of both sides with respect to time, t, we get:
dA/dt = 2πr dr/dt
where dr/dt is the rate of change of the radius.
We know that dA/dt = 5 cm²/s, and when the radius is 3 cm, we can plug in these values to solve for dr/dt:
5 = 2π(3) dr/dt
dr/dt = 5/(6π) cm/s ≈ 0.265 cm/s
Therefore, the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm.
b. To find how fast the radius is changing when the circumference is 1 cm, we can use the formula for the circumference of a circle:
C = 2πr
Taking the derivative of both sides with respect to time, t, we get:
dC/dt = 2π dr/dt
where dr/dt is the rate of change of the radius.
We know that when the circumference is 1 cm, C = 1 cm, so we can plug in these values to solve for dr/dt:
1 = 2π dr/dt
dr/dt = 1/(2π) cm/s ≈ 0.159 cm/s
Therefore, the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm.