The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 80 mm?

mm^3/s

12800pi

To find how fast the volume of a sphere is increasing, we will use the formula for the volume of a sphere:

V = (4/3)πr³

where V is the volume and r is the radius.

To find the rate at which the volume is increasing, we need to find dV/dt, the derivative of the volume with respect to time (t).

Let's first find an expression for the volume V in terms of the diameter d. The diameter is double the radius, so we have:

d = 2r

Solving for r, we get:

r = d/2

Substituting this expression for r into the formula for the volume of a sphere, we get:

V = (4/3)π(d/2)³

Simplifying, we have:

V = (4/3)π(d³/8)

To find dV/dt, we can take the derivative of V with respect to time t. Remember, we want to find how fast the volume is changing with respect to time.

dV/dt = d/dt [(4/3)π(d³/8)]

Now, let's find the derivative of V using the chain rule. Since d is a function of t, we can differentiate d and apply the chain rule to the whole expression:

dV/dt = (4/3)π * d/dt (d³/8)

To differentiate d³/8 with respect to t, we can use the power rule:

dV/dt = (4/3)π * (3d²/8) * (dd/dt)

Since we are given that the radius is increasing at a rate of 2 mm/s, we also know that the diameter is increasing at the same rate (since diameter = 2 * radius). So, dd/dt = 2 mm/s.

Now, we can substitute dd/dt = 2 into the expression:

dV/dt = (4/3)π * (3d²/8) * (2)

Simplifying further, we have:

dV/dt = (2/3)π * d²

Finally, plugging in the given diameter d = 80 mm, we can calculate the rate at which the volume is increasing:

dV/dt = (2/3)π * (80²) = (2/3)π * 6400

So, the volume is increasing at a rate of (2/3)π * 6400 mm³/s.

To find the rate at which the volume of the sphere is increasing, we need to use the volume formula for a sphere and differentiate it with respect to time.

The volume of a sphere can be calculated using the formula:

V = (4/3)πr^3,

where V is the volume and r is the radius.

First, we need to find the rate of change of the radius with respect to time. It is given that the radius is increasing at a rate of 2 mm/s. This means dr/dt = 2 mm/s.

Next, we need to express the diameter in terms of the radius. The diameter (d) is twice the radius (r), so d = 2r.

Given that the diameter is 80 mm, we can substitute this value into the equation above to find the radius:

80 = 2r

Solving for r, we get r = 40 mm.

Now we have the value of r and dr/dt, so we can substitute them into the formula for the derivative of volume:

V' = dV/dt = (dV/dr) * (dr/dt)

To find dV/dr, we differentiate the volume formula with respect to r:

dV/dr = (4/3)π * 3r^2

Substituting the value of r into the equation:

dV/dr = (4/3)π * 3(40)^2 = 6400π mm^3

Now we can calculate the rate at which the volume is increasing by substituting all the values we have obtained:

V' = (dV/dr) * (dr/dt) = 6400π mm^3 * 2 mm/s

Simplifying:

V' = 12800π mm^3/s

Therefore, the volume of the sphere is increasing at a rate of 12800π mm^3/s when the diameter is 80 mm.

V = (4/3)*pi*r^3

dV/dt = 4*pi*r^2*(dr/dt)

In this case, dr/dt = 2 mm/s and
r = 80 mm

Do the calculation for dV/dt