# Good ole hemispherical dome differentials problem

1. Nov 1, 2008

### mlschiff

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 0.03 cm thick to a hemispherical dome with diameter 54 m. (Round the answer to two decimal places.)

2. Relevant equations
V = 1/2(4/3(pi*r^3)) = 1/2(4/3(pi*(1/2D)^3)) = 1/2(4/3*pi(1/8D^3)) = 1/12pi*D^3

3. The attempt at a solution
How does V change if we change d from 54m to (54 = 0.03*10^-2)m?
dV/dD = 1/4*pi*D^2
dV = 1/4*pi*D^2*dD
= 1/4*pi*(54)^2(0.03*10^-2)

I got 0.679 as an awesome and got it wrong...

2. Nov 1, 2008

### HallsofIvy

Staff Emeritus
Why did you switch to diameter? Since you are only painting a hemisphere, the paint only increases the radius, not the diameter. And use $V= (2/3)\pi r^3$.

3. Nov 1, 2008

### mlschiff

Okay, using 2/3 seems a lot easier, but my professor gave an example in class of converting radius to diameter. Furthermore, I had a problem on a previous homework where I had to solve for the rate at which a sphere increased, and I got the right answer after converting the radius in the formula into diameter after attempting to solve for the rate at which the radius changed. If I do go about solving for the radius in this problem, what all needs to be done to address the problem of solving for diameter?

4. Nov 1, 2008

### Dick

You can use the diameter if you want. But after applying the paint the diameter becomes 54m+2*(0.03cm).

5. Nov 2, 2008

### mlschiff

okay. thanks for the help, all!

6. Mar 11, 2011

### CarolTaylor

Your original differential seems to be correct, you used sound mathematical principles, the only problem I see is the final answer. When I put .25*pi*(54)^2*.0003 I get .687. I have tried to figure out what you might have miss entered, but the formula you used was good.