Good ole hemispherical dome differentials problem

[mlschiff]

Homework Statement

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.)

Homework 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

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...

 

[HallsofIvy]

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

 

[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?

 

[Dick]

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

 

[mlschiff]

okay. thanks for the help, all!

 

[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.