Differentials and paint needed problem

[synergix]

Homework Statement

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

Homework Equations

A= 2(pi)r²

The Attempt at a Solution

dr=0.0005 m
r=25m
dA=?

(A=2(pi)r²)'

dA= 4(pi)r*dr

I won't 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.

 

[lanedance]

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

 

[HallsofIvy]

What is the formula for volume of a sphere?

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

 

[synergix]

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?

 

[lanedance]

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?

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

 

[lanedance]

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

 

[synergix]

dv=2pi*r^2

 

[lanedance]

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