To find the linear speed [tex]\( v \)[/tex] of a point traveling at a constant speed along the circumference of a circle, we use the relationship between linear speed [tex]\( v \)[/tex], radius [tex]\( r \)[/tex], and angular speed [tex]\( \omega \)[/tex]. The formula is given by:
[tex]\[ v = r \times \omega \][/tex]
Given the radius [tex]\( r \)[/tex] of the circle is [tex]\( 35 \)[/tex] inches and the angular speed [tex]\( \omega \)[/tex] is [tex]\( \frac{3\pi}{7} \)[/tex] radians per second, we substitute these values into the formula.
First, substituting the radius:
[tex]\[ r = 35 \text{ inches} \][/tex]
Next, substituting the angular speed:
[tex]\[ \omega = \frac{3\pi}{7} \text{ radians per second} \][/tex]
Now, we calculate the linear speed:
[tex]\[ v = 35 \text{ inches} \times \frac{3\pi}{7} \text{ radians per second} \][/tex]
Perform the multiplication:
[tex]\[ v = 35 \times \frac{3\pi}{7} \][/tex]
Simplify the expression by multiplying:
[tex]\[ v = \frac{35 \times 3\pi}{7} \][/tex]
[tex]\[ v = \frac{105\pi}{7} \][/tex]
Finally, simplify the division:
[tex]\[ v = 15 \pi \text{ inches per second} \][/tex]
Given this, the exact, fully simplified answer for the linear speed [tex]\( v \)[/tex] is:
[tex]\[ v = 15 \pi \text{ inches per second} \][/tex]
