# Differentials and paint needed problem

1. Oct 27, 2009

### synergix

1. The problem statement, all variables and given/known data

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50m

2. Relevant equations

A= 2(pi)r2

3. The attempt at a solution

dr=0.0005 m
r=25m
dA=?

(A=2(pi)r2)'

dA= 4(pi)r*dr

I wont go any further because the number is very small and i think incorrect.
I am pretty sure I'm missing a step but I can't figure out what or why.
Then again I could be way off.

2. Oct 28, 2009

### lanedance

Re: differentials

the amount of paint will be a volume, not area, think of a thin spherical shell

3. Oct 28, 2009

### HallsofIvy

Staff Emeritus
Re: differentials

What is the formula for volume of a sphere?

($\pi r^2$ is the area of circle and not relevant here.)

4. Oct 28, 2009

### synergix

Re: differentials

volume of a sphere= (4/3)pi*r^2

volume of hemisphere= (4/6)pi*r^2

dv= (4/3)pi*r*dr

dv= (4/3)*pi*25*.0005

am I wrong still?

5. Oct 28, 2009

### lanedance

Re: differentials

yes...

volume of a sphere is (4/3)pi*r^3

and when you differntiate a power, you multiply by the orgiginal power, not divide

6. Oct 28, 2009

### lanedance

Re: differentials

that looks like the whole sphere, how about the hemisphere part?

7. Oct 28, 2009

### synergix

Re: differentials

dv=2pi*r^2

8. Oct 28, 2009

### lanedance

Re: differentials

almost.... you just need to add a dr in there

so to sumamrise
V(r) = (1/2)(4/3)pi.r^3 = (2/3)pi.r^3

then the derivative is
dV/dr = 2pi*r^2

so for a small change in r, ∆r the approximate corresponding change in ∆V volume will be
∆V= (dV/dr).∆r = 2pi.r^2.∆r