(a) Answer The area of a circle as a function of the circumference C is A = C 2 4 π Explanation Given Information: The figure in discussion is a circle. Calculation: The area of the circle is given by, A = π r 2 ............... (1) Here, A is the area of the circle and r is the radius of the circle. The circumference of a circle is given by, C = 2 π r ............. (2) Here, C is the circumference of the circle and r is the radius of the circle. The value of r, from equation (2) in terms of circumference is, C = 2 π r r = C 2 π Substitute the value of r in equation (1), A = π r 2 = π ( C 2 π ) 2 = C 2 4 π Conclusion: The area of a circle as a function of the circumference C is A = C 2 4 π (b) To determine To calculate: The rate of change of area with respect to the circumference C. Answer The rate of change of area with respect to the circumference is C 2 π . Explanation Given Information: The area of a circle as a function of the circumference C is A = C 2 4 π Calculation: The area of a circle as a function of the circumference C is, A = C 2 4 π The rate of change of area is equal to the derivative of area with respect to C , A = C 2 4 π d A d C = d d C ( C 2 4 π ) = 2 C 4 π = C 2 π Here, d A d C is the rate of change of area with respect to the circumference. Conclusion: The rate of change of area with respect to the circumference is C 2 π . (c) To determine To calculate: The rate of change of area with respect to circumference C at C = π and C = 6 π Answer The rate of change of area with respect to the circumference at C = π is 1 2 . The rate of change of area with respect to the circumference at C = 6 π is 3 Explanation Given Information: The rate of change of area with respect to the circumference is, d d C A = C 2 π Calculation: The rate of change of area with respect to the circumference is, d d C A = C 2 π The rate of change of area with respect to the circumference at C = π is, [ d d C A ] C = π = [ C 2 π ] C = π = π 2 π = 1 2 Here, [ d V d s ] C = π is the rate of change of area with respect to the circumference at C = π . The rate of change of area with respect to the circumference at C = 6 π is, [ d d C A ] C = 6 π = [ C 2 π ] C = 6 π = 6 π 2 π = 3 Here, [ d V d s ] C = 6 π is the rate of change of area with respect to the circumference at C = 6 π . Conclusion: The rate of change of area with respect to the circumference at C = π is 1 2 . The rate of change of area with respect to the circumference at C = 6 π is 3 (d) To determine To calculate: The appropriate units for d A d C . Answer The appropriate unit for d A d C = in . Explanation Given Information: The rate of change of area with respect to the circumference. Calculation: C is measured in inches and A is measured in square inches. The rate of change of area with respect to the circumference is, d A d C Substitute the units in above equation, d A d C = in 2 in = in Conclusion: The appropriate unit for d A d C = in .

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 3.4, Problem 2E

(a)

To determine

To calculate: The area of a circle as a function of the circumference C.

(a)

Expert Solution

Answer to Problem 2E

The area of a circle as a function of the circumference C is A=C24π

Explanation of Solution

Given Information: The figure in discussion is a circle.

Calculation:

The area of the circle is given by,

A=πr2............... (1)

Here, A is the area of the circle and r is the radius of the circle.

The circumference of a circle is given by,

C=2πr ............. (2)

Here, C is the circumference of the circle and r is the radius of the circle.

The value of r, from equation (2) in terms of circumference is,

C=2πrr=C2π

Substitute the value of r in equation (1),

A=πr2=π(C2π)2=C24π

Conclusion:

The area of a circle as a function of the circumference C is A=C24π

(b)

To determine

To calculate: The rate of change of area with respect to the circumference C.

(b)

Expert Solution

Answer to Problem 2E

The rate of change of area with respect to the circumference is C2π.

Explanation of Solution

Given Information: The area of a circle as a function of the circumference C is A=C24π

Calculation:

The area of a circle as a function of the circumference C is,

A=C24π

The rate of change of area is equal to the derivative of area with respect to C ,

A=C24πdAdC=ddC(C24π)=2C4π=C2π

Here, dAdC is the rate of change of area with respect to the circumference.

Conclusion:

The rate of change of area with respect to the circumference is C2π.

(c)

To determine

To calculate: The rate of change of area with respect to circumference C at C=π and C=6π

(c)

Expert Solution

Answer to Problem 2E

The rate of change of area with respect to the circumference at C=π is 12.

The rate of change of area with respect to the circumference at C=6π is 3

Explanation of Solution

Given Information: The rate of change of area with respect to the circumference is,

ddCA=C2π

Calculation:

The rate of change of area with respect to the circumference is,

ddCA=C2π

The rate of change of area with respect to the circumference at C=π is,

[ddCA]C=π=[C2π]C=π=π2π=12

Here, [dVds]C=π is the rate of change of area with respect to the circumference at C=π .

The rate of change of area with respect to the circumference at C=6π is,

[ddCA]C=6π=[C2π]C=6π=6π2π=3

Here, [dVds]C=6π is the rate of change of area with respect to the circumference at C=6π .

Conclusion:

The rate of change of area with respect to the circumference at C=π is 12.

The rate of change of area with respect to the circumference at C=6π is 3

(d)

To determine

To calculate: The appropriate units for dAdC.

(d)

Expert Solution

Answer to Problem 2E

The appropriate unit for dAdC=in.

Explanation of Solution

Given Information: The rate of change of area with respect to the circumference.

Calculation: C is measured in inches and A is measured in square inches.

The rate of change of area with respect to the circumference is,

dAdC

Substitute the units in above equation,

dAdC=in2in=in

Conclusion:

The appropriate unit for dAdC=in.

Chapter 3.4, Problem 1E

Chapter 3.4, Problem 3E

Chapter 3 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 3.1 - Prob. 1QR Ch. 3.1 - Prob. 2QR Ch. 3.1 - Prob. 3QR Ch. 3.1 - Prob. 4QR Ch. 3.1 - Prob. 5QR Ch. 3.1 - Prob. 6QR Ch. 3.1 - Prob. 7QR Ch. 3.1 - Prob. 8QR Ch. 3.1 - Prob. 9QR Ch. 3.1 - Prob. 10QR

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