Chapter 3: Q. 22 (page 299)

In Exercises 19–26, write down an equation that relates the two quantities described. Then use implicit differentiation to obtain a relationship between the rates at which the quantities change over time.
The volume V and radius r of a cylinder with a fixed height of 10 units.

Short Answer

Expert verified

The equation that relates the volume Vand the radius r of the cylinder is $V = 10 {πr}^{2}$ .

The derivative $\frac{d V}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{d V}{d t} = 20 πr \frac{dr}{dt}$ .

Step by step solution

Step 1. Given information.

The height of a cylinder is 10 units.

Step 2. Formula used.

The volume of a cylinder is $V = {πr}^{2} h$ cu. units.

Step 3. Apply the value of h.

Apply the value of $h = 10$ in $V = {πr}^{2} h$ as follows.

$V = {πr}^{2} h V = {πr}^{2} (10) V = 10 {πr}^{2}$

The equation that relates the volumeV and radiusr of the cylinder is $V = 10 {πr}^{2}$ .

Step 4. Apply the differentiation.

Apply the differentiation to $V = 10 {πr}^{2}$ with respect to t as follows.

$\frac{d}{d t} (V) = \frac{d}{d t} (10 {πr}^{2}) \frac{d V}{d t} = 20 πr \frac{dr}{dt}$

The derivative $\frac{d V}{d t}$ and $\frac{d r}{d t}$ are related by.

Step 5. Conclusion.

The equation that relates the volume V and the radius r of the cylinder is $V = 10 {πr}^{2}$ .

The derivative $\frac{d V}{d t}$ and $\frac{d r}{d t}$ are related by $\frac{d V}{d t} = 20 πr \frac{dr}{dt}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Exercises 19–26, write down an equation that relates the two quantities described. Then use implicit differentiation to obtain a relationship between the rates at which the quantities change over time.
The volume V and radius r of a cylinder with a fixed height of 10 units.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Apply the value of h.

Step 4. Apply the differentiation.

Step 5. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercises 19–26, write down an equation that relates the two quantities described. Then use implicit differentiation to obtain a relationship between the rates at which the quantities change over time.The volume V and radius r of a cylinder with a fixed height of 10 units.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Formula used.

Step 3. Apply the value of h.

Step 4. Apply the differentiation.

Step 5. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Exercises 19–26, write down an equation that relates the two quantities described. Then use implicit differentiation to obtain a relationship between the rates at which the quantities change over time.
The volume V and radius r of a cylinder with a fixed height of 10 units.