Chapter 3: Q. 19 (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 area A and perimeter P of a square.

Short Answer

Expert verified

The equation that relates the area A and perimeter P of a square is $A = \frac{P^{2}}{16}$ .

The derivative $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by $\frac{d A}{d t} = \frac{P}{8} (\frac{d P}{d t})$ .

Step by step solution

Step 1. Formula used.

The area of a square is $A = a^{2}$ sq. units.

The perimeter of a square is $P = 4 a$ units.

The perimeter can be written as follows.

$P = 4 a a = \frac{P}{4}$

Step 2. Apply the value.

The area A and perimeter P of the square are related by applying $a = \frac{P}{4}$ in $A = a^{2}$ as follows.

$A = a^{2} A = {(\frac{P}{4})}^{2} A = \frac{P^{2}}{16}$

The equation that relates the areaAand perimeterPof the square is $A = \frac{P^{2}}{16}$ .

Step 3. Apply the differentiation.

Apply the differentiation to $A = \frac{P^{2}}{16}$ with respect to time t as follows.

The derivatives $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by role="math" localid="1648734800512" $\frac{d A}{d t} = \frac{P}{8} (\frac{d P}{d t})$ .

Step 4. Conclusion.

The equation that relates the area A and perimeter P of the square is $A = \frac{P^{2}}{16}$ .

The derivative $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by $\frac{d A}{d t} = \frac{P}{8} (\frac{d P}{d t})$ .

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 area A and perimeter P of a square.

Short Answer

Step by step solution

Step 1. Formula used.

Step 2. Apply the value.

Step 3. Apply the differentiation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Applied Mathematics

Calculus

Decision Maths

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 area A and perimeter P of a square.

Short Answer

Step by step solution

Step 1. Formula used.

Step 2. Apply the value.

Step 3. Apply the differentiation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Applied Mathematics

Calculus

Decision Maths

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 area A and perimeter P of a square.