As a balloon in the shape of a sphere is being blown up, the radius is increasing 1/pi inches per second. At what rate is the volume increasing when the radius is 1 inch?
I know that the volume of a sphere = 4/3(pi)r^3
I don't know what to do next.
Figure dV/dt
dV/dr *dr/dt= dV/dt
you know the expression for dv/dr (take the derivative of V(r), and you are given dr/dt as 1/pi per sec.
To find the rate at which the volume is increasing when the radius is 1 inch, we can use the chain rule of differentiation.
We know that the volume of a sphere is given by the formula V = (4/3)πr^3, where V is the volume and r is the radius.
To find dV/dt, the rate at which the volume is changing with respect to time, we need to differentiate V with respect to both r and t.
First, let's find dV/dr, the rate at which the volume changes with respect to the radius. We can do this by taking the derivative of V with respect to r:
dV/dr = d/dt[(4/3)πr^3]
Taking the derivative with respect to r gives:
dV/dr = (4/3)(3πr^2)
Simplifying, we have:
dV/dr = 4πr^2
Now that we have a expression for dV/dr, we can use the chain rule to find dV/dt. The chain rule states that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx.
In this case, y = V, u = r, and x = t. So we can write:
dV/dt = dV/dr * dr/dt
Substituting the given value for dr/dt (1/pi inches per second), we have:
dV/dt = (4πr^2) * (1/pi)
Simplifying, we get:
dV/dt = (4r^2)/pi
Now, when the radius is 1 inch (r = 1), we can substitute this value into the expression for dV/dt:
dV/dt = (4(1^2))/pi
Simplifying, we find:
dV/dt = 4/pi cubic inches per second
Therefore, when the radius is 1 inch, the volume is increasing at a rate of 4/pi cubic inches per second.
To find the rate at which the volume is increasing when the radius is 1 inch, we need to find the derivative of the volume formula with respect to time.
Given that the volume of a sphere is V = (4/3)πr^3, we can find the derivative of V with respect to r by differentiating the equation:
dV/dr = d/dt[(4/3)πr^3]
Differentiating using the power rule, we get:
dV/dr = (4/3)π * 3r^2
Simplifying further, we have:
dV/dr = 4πr^2
Now, we can use the chain rule to find dV/dt (the rate of change of volume with respect to time):
dV/dt = (dV/dr) * (dr/dt)
Substituting in the given value of dr/dt as 1/π inches per second, we have:
dV/dt = (4πr^2) * (1/π)
Simplifying further, we get:
dV/dt = 4r^2
Now, to find the rate at which the volume is increasing when the radius is 1 inch, we substitute the given radius value of r = 1 into the equation:
dV/dt = 4(1)^2
Simplifying further, we find:
dV/dt = 4
Therefore, the volume is increasing at a rate of 4 cubic inches per second when the radius is 1 inch.