A spherical balloon is leaking air at 2 cubic inches per hour. How fast is the balloon's radius changing when the radius is 2 inches.
To find how fast the balloon's radius is changing, we need to use the derivative. Specifically, we can use related rates, which is a method of finding the rate at which one quantity changes with respect to the rate at which another related quantity changes.
Let's denote the volume of the balloon as V and the radius as r. We are given that the volume is changing at a rate of -2 cubic inches per hour (∆V/∆t = -2). Note that the negative sign indicates that the volume is decreasing.
The volume of a spherical balloon can be calculated using the formula V = (4/3)πr³, where π (pi) is a mathematical constant approximately equal to 3.14159.
To find how fast the radius is changing (∆r/∆t), we can use the chain rule of differentiation. The chain rule states that if a variable depends on another variable, we can differentiate it with respect to the independent variable by multiplying the derivatives of each variable with respect to the independent variable.
Taking the derivative of both sides of the volume equation, we have:
dV/dt = d/dt[(4/3)πr³]
Using the power rule of differentiation, the derivative of r³ is 3r²:
dV/dt = (4/3)π(3r² * dr/dt)
Simplifying, we have:
-2 = (4/3)π(3r² * dr/dt)
The 3s cancel out, and we can solve for dr/dt:
dr/dt = (-2 * 3) / [(4/3)π(3r²)]
dr/dt = -6 / [4πr²]
Now we have an equation to find the rate at which the radius is changing. Plugging in the given radius (r = 2 inches):
dr/dt = -6 / [4π(2²)]
dr/dt = -6 / [16π]
Simplifying further:
dr/dt ≈ -0.119 inches per hour
Therefore, when the radius is 2 inches, the balloon's radius is decreasing at a rate of approximately 0.119 inches per hour.
To find the rate at which the balloon's radius is changing, we can use a derivative. Let's denote the radius of the balloon as r (in inches) and the volume of the balloon as V (in cubic inches).
We know that the volume of a sphere is given by the formula:
V = (4/3)πr^3,
where π is a constant (approximately equal to 3.14159). We can differentiate both sides of this equation with respect to time (t) since we are given the rate of change of the volume with respect to time.
dV/dt = d((4/3)πr^3)/dt.
Using the chain rule, d((4/3)πr^3)/dt can be written as:
dV/dt = (4/3)π * d(r^3)/dt.
To find d(r^3)/dt, we can differentiate r^3 with respect to t:
d(r^3)/dt = 3r^2 * dr/dt.
Now, substituting this back into our equation:
dV/dt = (4/3)π * (3r^2 * dr/dt).
We are given that dV/dt is equal to -2 cubic inches per hour, and we want to find dr/dt when r = 2 inches. So, let's substitute these values into our equation:
-2 = (4/3)π * (3(2)^2 * dr/dt).
Simplifying further:
-2 = (4/3)π * (12 * dr/dt),
-2 = 16π * dr/dt.
Now, let's solve for dr/dt:
dr/dt = -2/(16π),
dr/dt = -1/(8π).
Therefore, the rate at which the balloon's radius is changing when the radius is 2 inches is approximately -1/(8π) inches per hour.