The volume of a cylinder of height 11 inches and radius r inches is given by the formula V=11πr2 .
Suppose that the radius is expanding at a rate of 0.4 inches per second. How fast is the volume changing when the radius is 2.9 inches? Use at least 5 decimal places in your answer.
answer: ____ cubic inches per second
To find how fast the volume is changing with respect to time, we need to take the derivative of the volume function with respect to time.
Given: V = 11πr²
We can rewrite the volume equation as:
V = 11π * (r(t))² [r is a function of t]
To find dV/dt, we apply the chain rule:
dV/dt = d/dt (11π * r(t)²)
= 11π * d/dt (r(t)²)
= 11π * 2r(t) * (dr(t)/dt)
We are given that dr/dt = 0.4 inches per second, and we want to find dV/dt when the radius is 2.9 inches.
Substituting r = 2.9 and dr/dt = 0.4 into the equation, we have:
dV/dt = 11π * 2(2.9) * 0.4
Calculating this value:
dV/dt = 11π * 2 * 2.9 * 0.4
Using a calculator, we get:
dV/dt = 8.305308726 cubic inches per second
So, the volume is changing at a rate of approximately 8.3053 cubic inches per second when the radius is 2.9 inches.
To find how fast the volume is changing, we need to take the derivative of the volume with respect to time. Let's denote the radius as "r" (in inches) and time as "t" (in seconds). The volume of the cylinder is given by the equation V = 11πr^2.
To find dV/dt (the rate at which the volume is changing with respect to time), we apply the chain rule:
dV/dt = dV/dr * dr/dt
where dV/dr represents the derivative of V with respect to r, and dr/dt represents the rate at which the radius is changing with respect to time.
Taking the derivative of V with respect to r, we get:
dV/dr = 22πr
The rate at which the radius is changing, dr/dt, is given as 0.4 inches per second.
Now, we can calculate how fast the volume is changing when the radius is 2.9 inches:
V = 11π(2.9)^2
dV/dr = 22π(2.9)
dr/dt = 0.4
Now we can substitute these values into the equation:
dV/dt = (22π(2.9))(0.4)
Calculating this value, we get:
dV/dt ≈ 20.66158 cubic inches per second.
Therefore, when the radius is 2.9 inches, the volume is changing at a rate of approximately 20.66158 cubic inches per second.
V = pi R^2 h
dV/dR = pi h (2 R ) = 2 R pi h
so
dV/dt = h * (2 pi R) dR/dt
or in other words h times the circumference * dR /dt (sketch it :)