Good ole hemispherical dome differentials problem

In summary, this conversation discusses the use of differentials to estimate the amount of paint needed for a hemispherical dome with a diameter of 54m. The formula V = (2/3)*pi*r^3 is used to calculate the volume of paint needed, and the formula dV = (1/4)*pi*D^2*dD is used to calculate the rate of change in volume with respect to the diameter. There is a discrepancy in the final answer, but the principles used in the calculations are correct.
  • #1
mlschiff
4
0

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...
 
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  • #2
Why did you switch to diameter? Since you are only painting a hemisphere, the paint only increases the radius, not the diameter. And use [itex]V= (2/3)\pi r^3[/itex].
 
  • #3
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
You can use the diameter if you want. But after applying the paint the diameter becomes 54m+2*(0.03cm).
 
  • #5
okay. thanks for the help, all!
 
  • #6
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.
 

FAQ: Good ole hemispherical dome differentials problem

1. What is a good ole hemispherical dome differentials problem?

A good ole hemispherical dome differentials problem is a mathematical problem that involves finding the surface area and volume of a hemispherical dome when the top portion is cut off at a specific height.

2. What is the significance of studying this problem?

This problem is often used in real-world applications, such as in architecture and engineering, where the surface area and volume of hemispherical domes need to be calculated for construction and design purposes.

3. What are the key equations used to solve this problem?

The key equations used to solve this problem are the formula for the surface area of a sphere (4πr2) and the formula for the volume of a hemisphere (2/3πr3).

4. What are the possible variations of this problem?

The problem can be varied by changing the dimensions of the dome, such as the radius or height, or by altering the shape of the cut-off portion, such as using a conical or parabolic shape instead of a flat cut-off.

5. What are some tips for solving this problem efficiently?

One tip is to break down the problem into smaller parts, such as finding the surface area and volume separately, and then combining the results. It is also important to carefully label and keep track of all the given information and variables in order to avoid errors in calculation.

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