Question

find the linear speed v for the following. a point on the equator moving due to saturn’s rotation, if the radius is 36,180 mi and a rate is one revolution every 11 hours the linear speed v for a point on the equator moving due to saturn’s rotation, if the radius is 36,180 mi and a rate is one revolution every 11 hours is (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) mph.

Explanation:

Step1: Recall the formula for linear speed

The linear speed \( v \) of a point moving in a circular path is given by \( v=\frac{s}{t} \), where \( s \) is the arc length (distance traveled) and \( t \) is the time taken. For one revolution, the arc length \( s \) is the circumference of the circle, which is \( C = 2\pi r \), where \( r \) is the radius of the circle.

Step2: Calculate the circumference

Given the radius \( r = 36180 \) miles, the circumference \( C=2\pi r=2\pi\times36180 = 72360\pi \) miles.

Step3: Determine the time for one revolution

The time \( t \) for one revolution is 11 hours.

Step4: Calculate the linear speed

Using the formula \( v=\frac{s}{t} \), where \( s = 72360\pi \) miles and \( t = 11 \) hours, we have \( v=\frac{72360\pi}{11} \). Simplifying \( \frac{72360}{11}=6578\frac{2}{11} \)? Wait, no, \( 72360\div11 = 6578.1818\cdots \)? Wait, no, let's do the division: \( 11\times6578 = 72358 \), so \( 72360-72358 = 2 \), so \( \frac{72360}{11}=6578+\frac{2}{11}=\frac{6578\times11 + 2}{11}=\frac{72358+2}{11}=\frac{72360}{11} \). But actually, \( 72360\div11 = 6578.1818\cdots \) but we can keep it as a fraction. Wait, \( 72360 = 11\times6578 + 2 \), so \( \frac{72360\pi}{11}=\frac{(11\times6578 + 2)\pi}{11}=6578\pi+\frac{2\pi}{11} \)? No, that's not right. Wait, no, the formula is \( v=\frac{2\pi r}{t} \), where \( r = 36180 \), \( t = 11 \). So \( v=\frac{2\pi\times36180}{11}=\frac{72360\pi}{11} \). Let's simplify \( 72360\div11 \): 116578 = 72358, 72360 - 72358 = 2, so \( \frac{72360}{11}=\frac{72358 + 2}{11}=6578+\frac{2}{11}=\frac{6578\times11+2}{11}=\frac{72358 + 2}{11}=\frac{72360}{11} \). But maybe we can factor 72360 and 11: 11 is a prime number. 72360 divided by 11: 116000=66000, 72360 - 66000=6360; 11500=5500, 6360 - 5500=860; 1170=770, 860 - 770=90; 118=88, 90 - 88=2. So total is 6000+500+70+8=6578, remainder 2. So \( \frac{72360\pi}{11}=\frac{(11\times6578 + 2)\pi}{11}=6578\pi+\frac{2\pi}{11} \), but that's not simpler. Wait, maybe I made a mistake in the formula. Wait, linear speed for circular motion is also \( v = r\omega \), where \( \omega \) is the angular speed in radians per unit time. The angular speed for one revolution (which is \( 2\pi \) radians) in \( t \) hours is \( \omega=\frac{2\pi}{t} \) radians per hour. Then \( v = r\omega=r\times\frac{2\pi}{t}=\frac{2\pi r}{t} \), which is the same as before. So with \( r = 36180 \), \( t = 11 \), so \( v=\frac{2\times\pi\times36180}{11}=\frac{72360\pi}{11} \). Let's compute 72360 divided by 11: 72360 ÷ 11 = 6578.1818... but as an exact fraction, it's \( \frac{72360\pi}{11} \). Wait, but maybe we can simplify 72360 and 11: 11 is prime, 72360 is 72360. So the exact value is \( \frac{72360\pi}{11} \) mph. Let's check: 2pi*36180 = 72360 pi, divided by 11 hours, so speed is \( \frac{72360\pi}{11} \) miles per hour.

Answer:

\( \dfrac{72360\pi}{11} \)