# Optimization Problem Homework: Need Help Finding Expression for h

• niravana21
In summary, the problem is trying to find the derivative of an expression involving "H". The author has tried substituting back into the Volume function and A equals 1. However, the new expression is messy and he needs help. He's stuck at trying to find the derivative of the new expression, but with the help of his classmate, they are able to simplify the expression and get: V = r^2h. They are still having some difficulty with finding the derivative, but they think they've found it.
niravana21

## Homework Statement

Here is the exact problem, in order to avoid confusion

## The Attempt at a Solution

I know that I have to find an expression for "h" and substitute it back into the Volume, but the way I do it, it just becomes way too messy to find the derivative.

Any ideas?

What way do you do it?

well I tried to set the Area function equal to "H". Then I took that expression and substituted it back into the Volume function for "H".

Now the next step is to find the derivative, but I can't do that, since the new expression is messy. That's basically where I need the help.

Don't set A equal to h, set A equal to 1, then solve for h. You want to set A = 1 since you have constant surface area 1. Then you plug that expression for h into V. What you end up with there shouldn't be too bad.

sorry that's what I meant. But the expression for H in terms of R becomes complicated with the square root and quotient.

here's what I get for h in terms of r:

h=sqrt[(1/pi^2r^2) - (r^2)]

now I get stuck at trying to find the derivative of that.

right, but it might help to rewrite it:

\begin{align*} h &= \sqrt{\frac{1}{\pi^2 r^2} - r^2}\\ &= \sqrt{\frac{1}{\pi^2 r^2} - \frac{\pi^2 r^4}{\pi^2 r^2}}\\ &= \sqrt{\frac{1 - \pi^2r^4}{\pi^2r^2}}\\ &= \frac{\sqrt{1 - \pi^2r^4}}{\pi r} \end{align*}

So then
\begin{align*} V &= \frac{1}{3}\pi r^2 h\\ &= \frac{1}{3}\pi r^2 \frac{\sqrt{1 - \pi^2r^4}}{\pi r}\\ &= \frac{1}{3}r \sqrt{1 - \pi^2r^4} \end{align*}

Still a bit of a pain, but not as bad as before.

wow that does simply it! Now I find the derivative. But then would if I set the derivative equal to zero and find a critical point?

Also I'm having some difficulty of still finding the derivative :(
After simplification I get:
(-6πr^4+1-π^2 r^4)/(3√(1-π^2 r^4 ))

which I think is wrong since this online derivative calculator gave me this:
http://www.numberempire.com/cgi-bin/render2.cgi?\nocache%20\LARGE%20\frac{\partial%20f}{\partial%20x}%20%3D%20-{{3\%2C\pi^2\%2Cx^4-1}\over{3\%2C\sqrt{1-\pi\%2Cx^2}\%2C\sqrt{\pi\%2Cx^2%2B1}}}

Last edited by a moderator:
Hmm.. Here's what I've got:

\begin{align*} V &= \frac{1}{3}r\sqrt{1 - \pi^2r^4}\\ V' &= \frac{1}{3}\sqrt{1 - \pi^2r^4} + \frac{1}{3}r\frac{1}{2}(1 - \pi^2r^4)^{-1/2}(-4\pi^2r^4)\\ &= \frac{1}{3}\sqrt{1 - \pi^2r^4} - \frac{2\pi^2r^4}{3\sqrt{1 - \pi^2r^4}}\\ &= \frac{1 - \pi^2r^4 - 2\pi^2r^4}{3\sqrt{1 - \pi^2r^4}}\\ &= \frac{1 - 3\pi^2r^4}{3\sqrt{1 - \pi^2r^4}}\\ \end{align*}

This is the same thing the derivative calculator got, but it factored the denominator. It looks like you only made a minor mistake. Anyway, to find critical points, you want to know where the derivative is 0 or undefined. The derivative will be undefined when its denominator is 0, but in problems like these, the critical point you're looking for usually comes from setting the derivative equal to 0 and solving.

## What is an optimization problem?

An optimization problem is a mathematical problem that involves finding the best possible solution for a given situation. It typically involves maximizing or minimizing a certain quantity, subject to a set of constraints.

## Why do we need to solve optimization problems?

Optimization problems are important because they allow us to make the best decisions possible in various real-life situations. They are used in many fields such as engineering, economics, and computer science to improve efficiency and effectiveness.

## What is the process for solving an optimization problem?

The general process for solving an optimization problem involves identifying the objective function, determining the constraints, and using mathematical techniques such as differentiation and optimization algorithms to find the optimal solution.

## What is an expression for h?

In the context of an optimization problem, an expression for h refers to the mathematical equation or formula that represents the function that is being optimized. This expression can be derived from the given problem and is used to find the optimal value for h.

## Can someone help me find an expression for h?

Yes, there are various resources available to help with solving optimization problems and finding expressions for h. You can consult with a math tutor, use online tools and calculators, or seek guidance from a fellow scientist or researcher.

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