Question

The Earth completely rotates on its axis once every 24 hours.

A) How long does it take for it to rotate 310 degrees?

B) How long does it take to rotate 5 radians?

C) The diameter of the Earth is approximately 7920 miles. How far will a point on the equator rotate in 2 hours?

Show all work. Give answers to the nearest hundredth. Include the units in your response.

Answer 1

Part A). In 20 hr 24 minutes earth rotate 310°.

Part B). In 19 hr 6 minutes earth rotate 5 radians

Part C). 2072.4 miles point on equator rotates in 2 hours.

Explanation:

Degree that earth rotate in 24 hour = 360°

Number of radian that earth rotate in 24 hour = 2π radian

Part A).

Time taken to rotate 360° = 24 hours

Time taken to rotate 1° =
`(24)/(360)=(1)/(15)\,hours`

Time taken to rotate 310° =
`310*(1)/(15)=20(2)/(5)=20\,hr\:24\,minutes`

Part B).

Time taken to rotate 2π radian = 24 hours

Time taken to rotate 1 radian =
`(24)/(2\pi)=(12)/(\pi)\,hours`

Time taken to rotate 5 radian =
`5*(12)/(\pi)=(60)/(\pi)=19.1\.hr=19\,hr\:6\,minutes`

Part C).

Diameter of Earth = 7920 miles

Radius of earth, r = 3960 miles

Degree of rotation in 1 hours =
`(360)/(24)=15^(\circ)`

Degree of rotation in 2 hours , [tex\theta[/tex] = 15 × 2 = 30°

Length of the arc for angle 30° of circle with radius 3960 miles = Distance covered by point in 2 hours.

Length of the arc =
`(\theta)/(360^(\circ))*2\pi r=(30)/(360)*2*3.14*3960=2072.4\:miles`

Therefore, Part A). In 20 hr 24 minutes earth rotate 310°.

Part B). In 19 hr 6 minutes earth rotate 5 radians

Part C). 2072.4 miles point on equator rotates in 2 hours.

Answer 2

A)

62/3 = 20.67 hours

B)

60 hours

C)

2074.29 miles

Explanation:

If we assume the earth is a perfect circle, then in a complete rotation the earth covers 360 degrees or 2π radians.

A)

In 24 hours the earth rotates through an angle of 360 degrees. We are required to determine the duration it takes to rotate through 310 degrees. Let x be the duration it takes the earth to rotate through 310 degrees, then the following proportions hold;

(24/360) = (x/310)

solving for x;

x = (24/360) * 310 = 62/3 = 20.67 hours

B)

In 24 hours the earth rotates through an angle of 2π radians. We are required to determine the duration it takes to rotate through 5π radians. Let x be the duration it takes the earth to rotate through 5π radians, then the following proportions hold;

(24/2π radians) = (x/5π radians)

Solving for x;

x = (24/2π radians)*5π radians = 60 hours

C)

If the diameter of the earth is 7920 miles, then in 24 hours a point on the equator will rotate through a distance equal to the circumference of the Earth. Using the formula for the circumference of a circle we have;

circumference = 2*π*R = π*D

= 7920π miles

Therefore, the speed of the earth is approximately;

(7920π miles)/(24 hours) = 330π miles/hr

The distance covered by a point in 2 hours will thus be;

330π * 2 = 660π miles = 2074.29 miles