Question

the planet xyz travels about the star abc in an orbit that is almost circular. assume that the orbit is a circle with radius 83,000,000 mi. assume there are 24 hours in one day on planet xyz. (a) assume that xyz planet year is 368 days, and find the angle formed by xyzs movement in one day. (a) the angle formed by xyzs movement in one day is □ radian(s). (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) (b) give the angular speed in radians per hour. (c) find the linear speed of xyz in miles per hour.

Explanation:

Step1: Calculate angle in one - day

A full - circle is \(2\pi\) radians and the planet's year is 368 days. To find the angle formed in one day, we use the proportion. The angle \(\theta\) in one day is given by \(\theta=\frac{2\pi}{368}=\frac{\pi}{184}\) radians.

Step2: Calculate angular speed in radians per hour

There are 24 hours in a day. The angular speed \(\omega\) in radians per hour is \(\omega=\frac{\frac{\pi}{184}}{24}=\frac{\pi}{184\times24}=\frac{\pi}{4416}\) radians per hour.

Step3: Calculate linear speed

The formula for linear speed \(v = r\omega\), where \(r = 83000000\) miles and \(\omega=\frac{\pi}{4416}\) radians per hour. So \(v=83000000\times\frac{\pi}{4416}=\frac{83000000\pi}{4416}=\frac{20750000\pi}{1104}\) miles per hour.

Answer:

(a) \(\frac{\pi}{184}\)
(b) \(\frac{\pi}{4416}\)
(c) \(\frac{20750000\pi}{1104}\)