BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.11, Problem 43E

(a)

To determine

To find: The number of times more radiation energy per unit area emitted by the sun than the earth.

Expert Solution

Answer to Problem 43E

The radiation energy per unit area emitted by sun is greater than earth by factor of 160,000.

Explanation of Solution

Given:

The total energy per unit area emitted by a heated surface is directly proportional to fourth power of its absolute temperature.

Temperature of surface of the sun is 6000K.

Temperature of the surface of the earth is 300K.

Calculation:

The total radiation energy (E) emitted per unit surface area by the heated object is directly proportional to the fourth power of its absolute temperature.

That is,

EA=kT4 (1)

Here,

E is the total radiation energy emitted by the heated surface.

A is the surface area.

T is the absolute temperature of the heated surface.

k is the constant of proportionality.

To calculate Radiation energy (E1) emitted per unit surface area by the sun, substitute 1 for A and 6000 for T in equation (1).

E1(1)=k(6000)4E1=k(6000)4 (2)

To calculate Radiation energy (E2) emitted per unit surface area by the earth, substitute 1 for A and 300 for T in equation (1).

E2(1)=k(300)4E2=k(300)4 (3)

Divide equation (2) by equation (3).

E1E2=k(6000)4k(300)4E1E2=(6000300)4E1E2=(20)4E1E2=160000

From the above equation,

E1=160000E2

Thus, the radiation energy emitted per unit surface area is greater than earth by factor of 160,000.

(b)

To determine

To find: The number of times more total radiation emitted by the sun than the earth.

Expert Solution

Answer to Problem 43E

The total radiation energy emitted by the sun is greater than the earth by the factor of 1,930,670,340.

Explanation of Solution

Given:

Radius (r1) of the earth is 3960mi.

Radius (r2) of the sun is 435,000mi.

Temperature of surface of the sun is 6000K.

Temperature of the surface of the earth is 300K.

Calculation:

Consider both earth and the sun as sphere.

Surface area of the sphere is 4πr2.

Here, r is the radius of the sphere.

Use equation (1) and calculate total radiation energy emitted by the sun.

Substitute 4πr12 for A, 6000 for T in equation (1) and calculate total energy emitted by the sun.

ESun4π(r1)2=k(6000)4 (4)

Now, use equation (1) and calculate total radiation energy emitted by the earth

Substitute 4πr22 for A, 300 for T in equation (1) and calculate total energy emitted by the earth.

Eearth4π(r2)2=k(300)4 (5)

Divide equation (4) by equation (5) and simplify.

ESun4π(r1)2Eearth4π(r2)2=k(6000)4k(300)4ESunEearth(r22r12)=(6000300)4

Now, substitute 3960 for r2 and 435000 for r1 in above equation.

ESunEearth(396024350002)=(6000300)4ESunEearth(8.28×105)=160000ESunEearth=1600008.28×105ESunEearth=1932367150

Thus, the total energy emitted by the sun is greater than the earth by the factor of approximately 1,930,670,340.

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