Question

(I don't udnerstand, please help w/ explination as well)

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

A) How long does it take for it to rotate 225 degrees?

B) How long does it take to rotate 9π 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

A). 15 hours

B). 108 hours

C). 2074.28 miles

Explanation:

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

It means the Earth takes 24 hours to complete 360° or 2π radians

Per hour rotation of the Earth will be =
`\frac{\text{Angle rotated in one rotation}}{\text{Time taken for one rotation}}`

=
`(360)/(24)=15` degree per hour

or
`(2\pi )/(24)=(\pi )/(12)` radians per hour

Now we have to calculate the time taken in 225° rotation.

∵ In 15° rotation was time taken = 1 hour

∴ In 1° rotation time taken by the Earth =
`(1)/(15)`

∴ In 225° time spent by the Earth =
`((1)(225))/(15)=15` hours

B). ∵
`(\pi )/(12)` radians rotation was completed in the time = 1 hour

∴ 1 radian rotation was completed in time =
`(1)/((\pi )/(12))=(12)/(\pi )`

∴ 9π radians rotation will be completed in time =
`(12(9\pi ))/(\pi )= 108` hours

Therefore, time taken in 9π rotation will be 108 hours.

C). If the diameter of the earth is 7920 miles then we have to calculate angle of rotation of a point on equator in 2 hours.

Since Length of arc = radius × angle of rotation

Since angle of rotation in 1 hour =
`(\pi )/(12)` radians

So angle of rotation in 2 hours =
`(2\pi )/(12)=(\pi )/(6)`

Now we put these values in the formula

Length of arc =
`(7920)/(2)((\pi )/(6))=660\pi` miles

= 660(3.1428)

= 2074.28 miles

Answer 2

A) 15 hours

B) 108 hours

C) 2073.45 miles

Explanation:

The earth rotates fully 1 time in 24 hours. Fully rotate means that it goes through 360 degrees in 24 hours.

A)

For this, we can set up unit ratio to solve.

"If earth rotates 360 degrees in 24 hours, 225 degrees take how much time (let it be x)?"

`(360)/(24)=(225)/(x)\\360x=24*225\\360x=5400\\x=(5400)/(360)\\x=15`

So , it takes 15 hours.

B)

Here, the rotation is given in radian, NOT degrees. We know that 2π radians is 360 degrees, thus we can say:

"If earth rotates 2π radians in 24 hours, 9π radians take how much time (let it be y)?"

`(2\pi)/(24)=(9\pi)/(y)\\2\pi y=9\pi * 24\\2\pi y = 216\pi\\y=(216\pi)/(2\pi)\\y=108`

So, it take 108 hours.

C)

The point on the equator is on the "outside" of the earth. So we need to figure out the circumference of the earth, given diameter is approximately 7920.

Circumference formula is C = 2πr, where C is the circumference, r is the radius (half of diameter, which is 7920/2 = 3960)

Hence

C = 2πr = 2π(3960) = 7920π

Hence, is 24 hours, a point travels 7920π miles. 2 hours is 1/12th of 24 hours, so in 2 hours the point will travel 1/12th the distance is travels in 24 hours. So:

`(7920\pi)/(12)=2073.45`

Thus, it will travel 2073.45 miles in 2 hours.