A spherical balloon expands uniformly as it is inflated. The radius of the balloon in r meters at time t seconds. Find the radius, the volume, and the surface area of the balloon when the rates of increase of the surface area of the balloon and the radius are numerically equal
da/dt = 8πr dr/dt
If da/dt = dr/dt, then r = 1/8π
so, now you can find the area and volume.
Steve, what is dr/dt? how did you get 1/8pi?
To find the radius, volume, and surface area when the rates of increase of the surface area and radius are numerically equal, we need to set up the equations for these parameters.
Let's assume that at time t, the radius of the balloon is r meters. The volume V of a sphere with radius r can be given by the formula:
V = (4/3) * π * r^3
The surface area A of a sphere with radius r can be given by the formula:
A = 4 * π * r^2
Now, let's find the rates of increase for both the surface area and the radius. We'll assume that the rates are represented by the derivatives, denoted as dA/dt (rate of increase of surface area) and dr/dt (rate of increase of radius).
Since the balloon is expanding uniformly, it means that the rates of increase of the surface area and radius will be numerically equal. Therefore, we can set up the equation:
dA/dt = dr/dt
Differentiating the volume equation with respect to time, we can find an expression for dr/dt:
dV/dt = (4/3) * π * (3r^2) * dr/dt
Since the balloon is inflating uniformly, dV/dt represents the rate of change of volume, which is constant. Therefore, it implies that dr/dt is also constant.
Now, substitute in the equation dA/dt = dr/dt:
4 * π * r^2 = dr/dt
To find the radius, we need to solve this differential equation. Rearranging the equation, we have:
4 * π * r^2 = dr/dt
Divide both sides by r^2 to isolate dr/dt:
4 * π = (dr/dt) / r^2
Now, integrate both sides with respect to t:
∫ (4 * π) dt = ∫ (dr/dt) / r^2 dt
4 * π * t + C1 = ∫ (dr/r^2)
Integrating (dr/r^2) gives us:
4 * π * t + C1 = -1/r + C2
Combining the constants C1 and C2 into a single constant C3, we have:
4 * π * t = -1/r + C3
Now, rearrange the equation to solve for r:
1/r = -4 * π * t + C3
Taking the reciprocal of both sides:
r = 1/(-4 * π * t + C3)
This formula gives us the radius of the balloon at any given time t.
To find the volume and surface area, substitute the expression for r into the respective formulas:
V = (4/3) * π * r^3
A = 4 * π * r^2
Using the expression for r, plug it into the volume and surface area formulas to obtain the equations in terms of t.
This is the approach to finding the radius, volume, and surface area when the rates of increase of the surface area and radius are numerically equal.