- #1

- 287

- 0

V=4/3*pi*r^2

dv=4/3*pi*2r dr/dt

V=x^3

dv=3x^2

- Thread starter chaotixmonjuish
- Start date

- #1

- 287

- 0

V=4/3*pi*r^2

dv=4/3*pi*2r dr/dt

V=x^3

dv=3x^2

- #2

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 9,970

- 132

When does the volume expansion rate of the sphere equal the volume expansion rate of the cube?

Okay, let's start:

Sphere:

The radius r, as a function of time, is given by r(t)=2+2t

Thus, its volume V, as a function of time, is given by: [itex]V(t)=\frac{4\pi}{3}(2+2t)^{3}[/itex]

Cube:

The cube's side, as a function of time, is given by s(t)=5+t.

Thus, its volume v, as a function of time is [itex]v(t)=(5+t)^{3}[/itex]

We wish to find t so that:

[tex]\frac{dV}{dt}=\frac{dv}{dt}[/tex]

This is essentially the setup.

However, with the measures you've given, even at t=0, the volumetric expansion rate of the sphere is greater than that of the cube.

- #3

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

First clarify the problem. You say the sphere is "growing at 2 m/s". That's a "rate of change of distance". Do you mean the radius is increasing at 2 m/s? Or the diameter? You say the cube is "expanding at 1 m/s". Do you mean that the length of an edge is increasing at 1 m/s? Since you are clearly using rate of change of a length here, I am not happy with the use of the same terminology, "expanding at the same rate", to refer to (apparently) rate of increase of the volume.

V=4/3*pi*r^2

dv=4/3*pi*2r dr/dt

V=x^3

dv=3x^2

Second be careful of your formulas. The volume of a sphere is given by "[ITEX](4/3)\pi r^3[/itex]", not "[itex]r^2[/itex]". For the sphere, [itex]dV/dt= 4\pi r^2 dr/dt[/itex] not what you have. Assuming that "growing at 2 m/s" means the radius is increasing at that rate, you know that dr/dt= 2 (If it is diameter that is increasing at that rate, then dD/dt= 2 dr/dt= 2 so dr/dt= 1). For the cube, yes, V= x

- #4

- 287

- 0

I do give the original dimensions of the cube and the sphere (well I just gave the r as 2).

- #5

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 956

Again, what is the formula for volume of a sphere?

- Last Post

- Replies
- 10

- Views
- 2K

- Last Post

- Replies
- 9

- Views
- 5K

- Replies
- 1

- Views
- 3K

- Replies
- 54

- Views
- 2K

- Replies
- 2

- Views
- 825

- Last Post

- Replies
- 4

- Views
- 490

- Replies
- 1

- Views
- 4K

- Replies
- 5

- Views
- 6K

- Last Post

- Replies
- 4

- Views
- 5K

- Replies
- 1

- Views
- 2K