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Find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius [tex]\( r = 35 \)[/tex] inches and angular speed [tex]\( \omega = \frac{3\pi \text{ rad}}{7 \text{ sec}} \)[/tex].

Note: Enter the exact, fully simplified answer.

[tex]\[
\nu = \_\_\_ \text{ in/sec}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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