Let A be the area of a circle with radius r that is increasing in size with respect to time. If the rate of change of the area is 8 cm/s, find the rate of change of the radius when the radius is 3 cm.
To find the rate of change of the radius when the radius is 3 cm, we can use the relationship between the area and the radius of a circle.
The formula for the area of a circle is A = πr², where A represents the area and r represents the radius.
Given that the rate of change of the area is 8 cm/s, we can differentiate both sides of the formula with respect to time to find the relationship between the rate of change of the area and the rate of change of the radius.
dA/dt = d/dt (πr²)
Now, let's find the derivative of the area A with respect to time t. Since the radius r is also a function of time, we need to apply the chain rule.
dA/dt = d/dt (πr²) = 2πr (dr/dt)
The rate of change of the area with respect to time is equal to 2πr times the rate of change of the radius with respect to time.
Given that the rate of change of the area is 8 cm/s, we have:
8 cm²/s = 2πr (dr/dt)
To find the rate of change of the radius when the radius is 3 cm, substitute r = 3 cm into the equation above and solve for (dr/dt).
8 cm²/s = 2π(3 cm) (dr/dt)
Simplifying further:
8 cm²/s = 6π cm (dr/dt)
Now, isolate (dr/dt) by dividing both sides of the equation by 6π cm:
(dr/dt) = (8 cm²/s) / (6π cm)
(dr/dt) ≈ 0.42 cm/s
Therefore, the rate of change of the radius when the radius is 3 cm is approximately 0.42 cm/s.
To find the rate of change of the radius, we need to differentiate the formula for the area of a circle with respect to time.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. Differentiating both sides of this equation with respect to time (t) yields:
dA/dt = 2πr dr/dt
where dA/dt is the rate of change of the area and dr/dt is the rate of change of the radius.
Given that dA/dt = 8 cm/s, and we want to find dr/dt when r = 3 cm, we can substitute these values into the equation:
8 = 2π(3) dr/dt
Now, we can solve for dr/dt:
dr/dt = 8 / (2π(3))
= 4 / (3π)
So, when the radius is 3 cm, the rate of change of the radius is 4 / (3π) cm/s.
ah, a nice related-rates problem.
a = pi r^2
da/dt = 2pi r dr/dt
when r=3,
8 = 2pi(3)(dr/dt)
dr/dt = 4/(3pi) cm/s
By the way, da/dt is 8 cm^2/s. That makes the units come out right.
cm^2/s = 2pi(cm)(cm/s)