Related Rates III 

A Conical Sand-Pile Problem 

Copyright Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

 

The steps in the document can be repeated to solve similar problems. 

Problem Statement 

Sand pouring from a hopper at a steady rate forms a conical pile whose height is observed to remain twice the radius of the base of the cone.  When the height of the pile is observed to be 20 feet, the radius of the base of the pile appears to be increasing at the rate of a foot every two minutes.  How fast is the sand pouring from the hopper? 

Solution 

 

Given :   when   To Find:

Table 1: Data stated in Problem Statement.

 

Write an equation for the volume of the cone, showing explicitly the dependence on .

 

(3.1)

 

Differentiate with respect to .

To differentiate the function, right click on the output and choose Differentiate>t

 

(3.2)

 

 

 

Table 1 provides the following information: 

 

 

Drag a copy of the differentiated equation to a new line, and make the appropriate substitutions as per Table 1.

Highlight the answer, [Ctrl]-drag the differentiated equation to a new line. Highlight each term and replace with its relevant value.

 

(3.3)

 

The sand is pouring from the hopper at a rate of . 

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