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