Calculus Question
- Thread starter henryjo
- Start date
- Joined
- Apr 12, 2005
- Messages
- 11,339
Spherical: Volume(Radius) = (4/3)*pi*Radius^3henryjo said:
dVolume = 4*pi*Radius^2*dr
Plug and chug.
"volume increases at a rate of 100cm^3/sec"
This gives dVolume = 100 cm^3/sec
"diameter is 50cm"
This gives Radius = 25 cm
100 cm^3/sec = 4*pi*(25 cm)^2*dr
Solve for 'dr'.
Now, what is the surface area of a sphere?
SurfaceArea(Radius) = 4*pi*Radius^2
dSurfaceArea = 8*pi*Radius*dr
Plug in what you know for 'Radius' and 'dr' and you are done.
Note: When you found the derivaite of the volume and out popped the formula for the surface area, a flag should have gone off in your head. "Hey, that's weird!" It is NOT a coincidence. It's just something to ponder as you wander through calculus. Don't lose any sleep over it. It doesn't help much in this problem.
One last part...
Okay, there is one last part I am struggling with:
I need to write an equation connecting the volume and surface area formula and find the partial derivative of the volume with respect to the surface area at the given diameter of the balloon (50 cm)
Okay, there is one last part I am struggling with:
I need to write an equation connecting the volume and surface area formula and find the partial derivative of the volume with respect to the surface area at the given diameter of the balloon (50 cm)
- Joined
- Apr 12, 2005
- Messages
- 11,339
7.999999999997 what? It's Surface Area. Perhaps cm^2?
V(r) = (4/3)*pi*r^3
dV = 4*pi*r^2*dr
SA(r) = 4*pi*r^2
r^2 = SA/(4*pi)
dSA = 8*pi*dr
dr = dSA/(8*pi)
dV = 4*pi*r^2*dr
dV = 4*pi*[SA/(4*pi)]*[dSA/(8*pi)]
dV = SA*[dSA/(8*pi)]
I guess. Seems like an awfully tough way to go about it.
V(r) = (4/3)*pi*r^3
dV = 4*pi*r^2*dr
SA(r) = 4*pi*r^2
r^2 = SA/(4*pi)
dSA = 8*pi*dr
dr = dSA/(8*pi)
dV = 4*pi*r^2*dr
dV = 4*pi*[SA/(4*pi)]*[dSA/(8*pi)]
dV = SA*[dSA/(8*pi)]
I guess. Seems like an awfully tough way to go about it.