# 1. You are to design a cylinder that will hold 50 cubic meters. What should the dimensions be to...

## Question:

1. You are to design a cylinder that will hold 50 cubic meters. What should the dimensions be to minimize the amount of material used?

2. You are to design a rectangular storage container with a volume of 50 cubic meters. If the base is a square, what should the dimensions be to minimize the amount of material?

## Maxima and minima using derivatives

Let {eq}\displaystyle r {/eq} be the radius and {eq}\displaystyle h {/eq} be the height of cylinder

Then volume,{eq}\displaystyle v=\pi *r^2h {/eq}

surface area {eq}\displaystyle s=2\pi*r^2+2\pi*r*h {/eq}

Let {eq}\displaystyle l {/eq} be the length {eq}\displaystyle b {/eq} be the width and {eq}\displaystyle h {/eq} be the height of rectangular box.

Then volume,{eq}\displaystyle v=lbh {/eq}

surface area {eq}\displaystyle s=2lb+2bh+2lh {/eq}

If second derivative of a function is greater than zero then equating first derivative to zero gives condition at which the functions attains minimum value.

And if second derivative is less than zero equating first derivative tozero gives condition for at which function attains maximum value.

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Let,

{eq}\displaystyle r {/eq} be the radius

{eq}\displaystyle h {/eq} be the height of cylinder

Given

Volume,

{eq}\displaystyle\begin{align} ... Solving Min-Max Problems Using Derivatives

from

Chapter 15 / Lesson 1
19K

Max and min problems show up in our daily lives extremely often. In this lesson, we will look at how to use derivatives to find maxima and minima of functions, and in the process solve problems involving maxima and minima.