Q. 20 In Exercises 19鈥�26, write down... [FREE SOLUTION]

Chapter 3: Q. 20 (page 299)

In Exercises 19鈥�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 an equilateral triangle.

Short Answer

Expert verified

The equation that relates the area A and perimeter Pof an equilateral triangle is $A = \frac{1}{12 \sqrt{3}} P^{2}$ .

The derivative $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by $\frac{d A}{d t} = \frac{P}{6 \sqrt{3}} \frac{d P}{d t}$ .

Step by step solution

Step 1. Formula used.

The area of an equilateral triangle is $A = \frac{\sqrt{3}}{4} a^{2}$ sq. units.

The perimeter of an equilateral triangle is $P = 3 a$ units.

The perimeter equation can also be written as follows.

$P = 3 a a = \frac{P}{3}$

Step 2. Apply the value.

Apply the value $a = \frac{P}{3}$ in $A = \frac{\sqrt{3}}{4} a^{2}$ as follows.

$A = \frac{\sqrt{3}}{4} a^{2} A = \frac{\sqrt{3}}{4} {(\frac{P}{3})}^{2} A = \frac{\sqrt{3}}{4} (\frac{P^{2}}{9}) A = \frac{1}{12 \sqrt{3}} P^{2}$

The equation that relates the area Aand the perimeter P of an equilateral triangle is $A = \frac{\sqrt{3}}{4} (\frac{P^{2}}{9})$ .

Step 3. Apply the differentiation.

Apply the differentiation to $A = \frac{1}{12 \sqrt{3}} P^{2}$ with respect to time t as follows.

$\frac{d}{d t} (A) = \frac{d}{d t} (\frac{1}{12 \sqrt{3}} P^{2}) \frac{d A}{d t} = (\frac{2 P}{12 \sqrt{3}}) \frac{d P}{d t} \frac{d A}{d t} = (\frac{P}{6 \sqrt{3}}) \frac{d P}{d t}$

The derivative $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by $\frac{d A}{d t} = \frac{P}{6 \sqrt{3}} \frac{d P}{d t}$ .

Step 4. Conclusion.

The equation that relates the area Aand perimeter P of an equilateral triangle is $A = \frac{1}{12 \sqrt{3}} P^{2}$ .

The derivative $\frac{d A}{d t}$ and $\frac{d P}{d t}$ are related by $\frac{d A}{d t} = \frac{P}{6 \sqrt{3}} \frac{d P}{d t}$ .

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In Exercises 19鈥�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 an equilateral triangle.

Short Answer

Step by step solution

Step 1. Formula used.

Step 2. Apply the value.

Step 3. Apply the differentiation.

Step 4. Conclusion.

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91影视

In Exercises 19鈥�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 an equilateral triangle.

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

Geometry

Calculus

Mechanics Maths

Probability and Statistics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 19鈥�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 an equilateral triangle.