To determine the linear speed [tex]\( v \)[/tex] of a point traveling in a circular motion, we can use the relationship between the arc length [tex]\( s \)[/tex], the time [tex]\( t \)[/tex], and the linear speed [tex]\( v \)[/tex]. The linear speed is given by the formula:
[tex]\[ v = \frac{s}{t} \][/tex]
where:
- [tex]\( s \)[/tex] is the arc length
- [tex]\( t \)[/tex] is the time taken to travel the arc length
Given:
- Arc length [tex]\( s = \frac{1}{15} \)[/tex] inches
- Time [tex]\( t = 8 \)[/tex] minutes
Substitute the given values into the formula:
[tex]\[ v = \frac{1/15 \text{ inches}}{8 \text{ minutes}} \][/tex]
Perform the division:
[tex]\[ v = \frac{1}{15} \div 8 = \frac{1}{15} \times \frac{1}{8} \][/tex]
[tex]\[ v = \frac{1}{15 \times 8} = \frac{1}{120} \][/tex]
So, the linear speed [tex]\( v \)[/tex] is:
[tex]\[ v = \frac{1}{120} \text{ inches per minute} \][/tex]
Converting the fraction to decimal form, we get:
[tex]\[ v = 0.008333333333333333 \text{ inches per minute} \][/tex]
Therefore, the exact linear speed is:
[tex]\[ v = \frac{1}{120} \text{ inches per minute} \][/tex]
And in decimal form, the linear speed is approximately:
[tex]\[ v = 0.008333333333333333 \text{ inches per minute} \][/tex]
