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17Calculus Derivatives - Related Rates Involving Cones

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This page covers related rates problems specifically involving volumes where the shape of the volume is a cone.

Overview

When solving these types of problems, you first draw of picture and pick out the type of geometric figure involved. The key here is to write down as many equations as you can think of and then use only the ones that seem to apply. Sometimes, especially with cones, you need to come up with relationships that might not be very intuitive. We will show you some things to watch for. Once you have a figure with all the parts labeled, you can write down the equations involved.

On the main related rates page, we discuss two ways to work these problems based on when you take the derivative, either before combining equations or after. Especially with volume problems, it is often better NOT to combine equations too soon, since the equations can get quite messy very quickly.

What To Do With Constants In Related Rates Problems

What do you do with constants that are given in the problem? First of all, you never want to just go in and plug in all your constants before you take the derivative.

Safe Answer - - Wait and plug in your constants only after you have the derivative. So, you would label all distances with variables, take the derivative with respect to t and then plug in all your given constants. This is what you need to do when you first start learning to work related rates problems. After you have some experience, you can go on to the more experienced technique.

Experienced Answer - - Once you learn the basics of related rates problems, you will have a feel for which constants you can plug in right away and which ones you can't. The difference you need to look for is
- if the variable is NOT changing, then you can substitute the constant in before taking the derivative;
- but, if the variable is changing over time, then you must wait until after you take the derivative before you can substitute the constant into the equation.

At this point, it will just confuse you more if we write down a bunch of theory on how to work these problems. You need to actually see one, then work many in order to see patterns. If you haven't already, read the top section (above the resources section) of the main related rates page. Then, come back here, watch a video or two and try your hand at a few problems.

Cones

Cone-shaped tanks are very interesting and you will probably run across more than one problem with a tank shaped like a cone. Before we get into the details of solving these kind of problems, here is a great video going through an example that the instructor explains very well.

AF Math & Engineering - Step by Step Method of Solving Related Rates Problems - Conical Example

There are several unique things that you need to understand and watch for when you work related rates problems that involve cones.
1. The formula for the volume of a cone with top radius \(r\) and height \(h\) is \(V = \pi r^2 h/3\).

2. There is a unique relationship that you may not think of when you work these problems. The idea is to take a cross-section of the cone down through the center to the tip of the cone. You end up with a triangle, actually two similar triangles.
- The larger triangle is the size of the tank.
- The smaller triangle is the size of the fluid in the tank.
Check the similar triangles precalculus page for a reminder on how to set up ratios of similar triangles.

3. One trick you need to watch for is whether the cone is point up or point down and what the problem is asking for. If the cone is point down, usually the fluid or whatever is filling the tank is in the shape of a cone, so there is nothing unusual going on IF they are asking for the volume of the fluid. However, if the cone is point up, you need to look at the problem carefully and think about what is going on. As time passes, the shape may or may not stay in the form of an exact cone. Here are a couple of examples.
- If you have something like gravel in a pile, this stays in a cone shape as gravel is being added to the pile. Nothing unusual is going on.
- One trick that I have seen on exams is, if something (like a fluid) is being removed from a cone-shape tank when the point is up, there will be some fluid at the bottom and no fluid at the tip (picture on the right). To calculate the volume of the fluid, you need to calculate the volume of the entire cone and subtract the volume of the top cone that is empty. (There are formulas for this volume but what do you do if you don't have the formula on the exam?)

Okay, time for a video. Here is an introduction video for cone problems. The writing is kind of hard to see but I think the discussion will help you get your head around what is going on. I particularly like how he describes the use of the chain rule to calculate the derivative.

Khan Academy - Introduction to rate-of-change problems [9min-34secs]

video by Khan Academy

Practice

A water tank has the shape of an inverted (point down) circular cone with base radius 2m and height of 4m. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the water is 3m deep.

Problem Statement

A water tank has the shape of an inverted (point down) circular cone with base radius 2m and height of 4m. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the water is 3m deep.

Solution

blackpenredpen - 3735 video solution

video by blackpenredpen

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Gravel is being dumped at a rate of 20ft3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high?

Problem Statement

Gravel is being dumped at a rate of 20ft3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high?

Solution

PatrickJMT - 1119 video solution

video by PatrickJMT

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A 10cm tall funnel (point down) is being drained with water at a constant rate of 10cc per second. The mouth of the funnel is 12cm in diameter. How fast is the water level dropping when there are 200cc left in the funnel? [cc = cubic centimeters]

Problem Statement

A 10cm tall funnel (point down) is being drained with water at a constant rate of 10cc per second. The mouth of the funnel is 12cm in diameter. How fast is the water level dropping when there are 200cc left in the funnel? [cc = cubic centimeters]

Solution

CalculusSuccess - 1120 video solution

video by CalculusSuccess

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A funnel is being drained of water at a rate of 0.1ft3/sec. How fast is the water level dropping when there is 4ft of water in the funnel? The full height of the funnel is 9ft and the radius at the top is 3ft.

Problem Statement

A funnel is being drained of water at a rate of 0.1ft3/sec. How fast is the water level dropping when there is 4ft of water in the funnel? The full height of the funnel is 9ft and the radius at the top is 3ft.

Final Answer

\( 9/(160\pi) \approx 0.0179 \) ft/sec

Problem Statement

A funnel is being drained of water at a rate of 0.1ft3/sec. How fast is the water level dropping when there is 4ft of water in the funnel? The full height of the funnel is 9ft and the radius at the top is 3ft.

Solution

Michel vanBiezen - 2246 video solution

video by Michel vanBiezen

Final Answer

\( 9/(160\pi) \approx 0.0179 \) ft/sec

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Gravel is being dumped from a conveyer belt at a rate of 100ft3/min forming a conical pile whose base diameter is two times the altitude. How fast is the height changing when the pile is 12ft high?

Problem Statement

Gravel is being dumped from a conveyer belt at a rate of 100ft3/min forming a conical pile whose base diameter is two times the altitude. How fast is the height changing when the pile is 12ft high?

Solution

The Organic Chemistry Tutor - 3713 video solution

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Water is leaking out of a conical tank at 500cm3/min. The tank has a height of 24cm and a radius of 6cm. Find the rate at which water is being poured into the tank if the water level is rising at a rate of 15cm/min when the height of the water is 9cm.

Problem Statement

Water is leaking out of a conical tank at 500cm3/min. The tank has a height of 24cm and a radius of 6cm. Find the rate at which water is being poured into the tank if the water level is rising at a rate of 15cm/min when the height of the water is 9cm.

Solution

The Organic Chemistry Tutor - 3714 video solution

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Really UNDERSTAND Calculus

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Topics You Need To Understand For This Page

derivatives

chain rule

implicit differentiation

basics of related rates

precalculus: word problems

For related rates problems involving similar triangles, it may help you to review how to set up the ratios. You can find a discussion of this on the similar triangles precalculus page.

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