When the height of a cylinder is 12 cm and the radius is 4 cm, the circumference of the cylinder is increasing at a rate of pie/4 cm/m in, and the height of the cylinder is increasing four times faster than the radius. How fast is the volume of the cylinder changing?

Question

When the height of a cylinder is 12 cm and the radius is 4 cm, the circumference of the cylinder is increasing at a rate of pie/4 cm/m in, and the height of the cylinder is increasing four times faster than the radius. How fast is the volume of the cylinder changing?

Answer 1

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Answer 2

since C = 2πr, dC/dt = 2π dr/dt

Since dC/dt = π/4, dr/dt = (π/4)/(2π) = 1/8 cm/min

v = πr^2h
dv/dt = 2πrh dr/dt + πr^2 dh/dt
= 2π*4*12*(1/8) + π*16*(1/2)
= 20π cm^3/min

Answer 3

Well, well, well! Let's calculate the rate of change of the volume of this hilarious cylinder, shall we?

Firstly, we need to find the rate at which the radius is changing. If the height is increasing four times faster than the radius, and the circumference is increasing at a constant rate of π/4 cm/m in, we can conclude that the radius is increasing at a rate of (π/4)/4 = π/16 cm/m.

Now, let's proceed to calculate the rate of change of the volume. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.

We know that h = 12 cm and r = 4 cm. Differentiating the equation with respect to time t, we get:

dV/dt = π(2rh)(dh/dt) + πr^2(dh/dt)

Substituting the values we have, we get:

dV/dt = π(2(4)(12))(4(π/16)) + π(4^2)(4(π/16))

Simplifying this circus of equations, we get:

dV/dt ≈ 96π(π/16) + 16π^2(π/16)
dV/dt ≈ 6π^2 + π^3/4 cm^3/m

So, the volume of this cylinder is changing at a speed of 6π^2 + π^3/4 cm^3/m. Isn't math magical? Well, maybe not as magical as a clown juggling bowling balls, but still pretty amusing!

Answer 4

To find how fast the volume of the cylinder is changing, let's first find the rates at which both the height and radius are changing.

Given:
Height (h) = 12 cm
Radius (r) = 4 cm
Circumference rate (dc/dt) = π/4 (cm/m in)
Height rate (dh/dt) = 4 * dr/dt

We need to find the Volume rate (dV/dt).

The formula for the volume of a cylinder is:
V = π * r^2 * h

First, let's find the rate at which the radius (r) is changing (dr/dt):
We are given that the circumference rate (dc/dr) is π/4 (cm/m in).
The formula for circumference is:
C = 2 * π * r

Differentiating both sides with respect to time (t), we get:
dc/dt = 2 * π * (dr/dt)

Since dc/dt = π/4 (cm/m in), we have:
π/4 = 2 * π * (dr/dt)
dr/dt = (π/4) / (2 * π)
dr/dt = 1/8 cm/m in

Now, let's find the rate at which the height (h) is changing (dh/dt):
We are given that the height rate (dh/dt) = 4 * dr/dt
Substituting the value of dr/dt = 1/8 cm/m in, we have:
dh/dt = 4 * (1/8) = 1/2 cm/m in

Now, let's find the rate at which the volume (V) is changing (dV/dt):
We can differentiate the volume formula with respect to time (t):
dV/dt = d/dt (π * r^2 * h)

Let's substitute the given values:
dV/dt = d/dt (π * (4^2) * 12)

Differentiating each term with respect to time (t):
dV/dt = d/dt (π * 16 * 12)
dV/dt = 16 * (d/dt (π * 12))
dV/dt = 16 * π * (dh/dt)

Substituting the value of dh/dt = 1/2 cm/m in:
dV/dt = 16 * π * (1/2)
dV/dt = 8π

Therefore, the volume of the cylinder is changing at a rate of 8π cm³/m in.

Answer 5

To find how fast the volume of the cylinder is changing, we need to use the formulas for the volume and rates of change.

The volume of a cylinder is given by the formula:
V = π * r^2 * h

Where:
V = Volume of the cylinder
π = Pi (approximately 3.14159)
r = Radius of the cylinder
h = Height of the cylinder

Now, let's differentiate the volume formula with respect to time to find the rate of change of the volume (dV/dt):

dV/dt = d/dt (π * r^2 * h)

To simplify this, we need to know the relationship between the height (h) and the radius (r).

Given that the height is increasing four times faster than the radius, we can express this relationship as:

dh/dt = 4 * dr/dt

Now, let's substitute this relationship into the volume differentiation:

dV/dt = d/dt (π * r^2 * h)
dV/dt = π * (2r * dr/dt * h + r^2 * dh/dt)

Since we have the values for h, r, and dh/dt from the problem statement, we can substitute them into the equation:

dV/dt = π * (2 * 4 * (π/4 cm/m) * 12 cm + (4 cm)^2 * 4 * (π/4 cm/m))

Now, let's simplify and calculate the value:

dV/dt = π * (8π cm^2/m + 16π cm^2/m)
dV/dt = π * (24π cm^2/m)

Therefore, the rate at which the volume of the cylinder is changing is 24π cm^3/m.