The volume of a cylinder increases at a rate of 324cm cube per second. The height, h cm of the cylinder is twice the radius, r cm. Find the rate of change in radius when the radius is 6cm.(use 3over2 pai)
v = πr^2h
dv/dt = 2πrh dr/dt + πr^2 dh/dt
Now just plug in your numbers to get dr/dt
Note that since h=2r, dh/dt = 2 dr/dt
Or, since h=2r,
v = πr^2h = 2πr^3
dv/dt = 6πr^2 dr/dt
To find the rate of change in radius when the radius is 6cm, we need to use the given information and apply the chain rule of differentiation.
Let's denote the volume of the cylinder as V, the radius as r, and the height as h. From the problem, we know that the volume is increasing at a rate of 324 cm^3 per second:
dV/dt = 324 cm^3/s (equation 1)
We are also given that the height, h cm, of the cylinder is twice the radius:
h = 2r (equation 2)
We are asked to find the rate of change in radius, dr/dt, when the radius is 6 cm.
To relate the volume to the radius and height, we need to use the formula for the volume of a cylinder, which is V = πr^2h.
Taking the derivative of both sides of this equation with respect to time (t), using the chain rule, we get:
dV/dt = d(πr^2h)/dt
Now let's differentiate each term separately:
dV/dt = 2πrh(dr/dt) + πr^2(dh/dt) (equation 3)
Since we know that the height, h, is equal to 2r, we can substitute this into equation 3:
dV/dt = 2πr(2r)(dr/dt) + πr^2(dh/dt)
dV/dt = 4πr^2(dr/dt) + 2πr^2(dh/dt)
We can simplify this further by substituting the given value of dV/dt:
324 = 4πr^2(dr/dt) + 2πr^2(dh/dt) (equation 4)
We need to find the value of dr/dt when r = 6 cm.
From equation 2, we know that h = 2r. By differentiating both sides with respect to t (time):
dh/dt = d(2r)/dt
dh/dt = 2(dr/dt) (equation 5)
Now, let's substitute the known values into equation 4 and equation 5:
324 = 4π(6^2)(dr/dt) + 2π(6^2)(2(dr/dt))
324 = 4π(36)(dr/dt) + 2π(36)(2(dr/dt))
324 = 4π(36)(dr/dt) + 72π(36)(dr/dt)
324 = (4π(36) + 72π(36))(dr/dt)
324 = (144π + 2592π)(dr/dt)
324 = 2736π(dr/dt)
Now, we can solve for dr/dt by rearranging the equation:
dr/dt = 324 / (2736π)
Using the value of 3.14 for π (approximately), we can evaluate dr/dt:
dr/dt = 324 / (2736 × 3.14)
dr/dt ≈ 0.0376 cm/s
Therefore, the rate of change in the radius when the radius is 6 cm is approximately 0.0376 cm/s.