Suppose that point P is on a circle with radius r, and ray OP is rotating with angular speed (n) Complete parts (a) through (c) r=14cm,omega =(pi )/(4)radian per sec, t=4sec (a) What is the angle generated by P in time t? Theta =square radian (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.) (b) What is the distance traveled by P along the circle in time t? s=square cm (Simplify your answer. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression.) (c) What is the linear speed of P? v= cm per sec

## Step 1<br />The problem involves the calculation of angular displacement, arc length, and linear speed. The formulas used to solve these problems are:<br /><br />### **Angular Displacement (\( \Theta \))**: <br />This is the angle through which point P rotates in time \( t \). It is calculated using the formula \( \Theta = \omega t \), where \( \omega \) is the angular speed and \( t \) is the time.<br /><br />### **Arc Length (\( s \))**: <br />This is the distance traveled by P along the circle in time \( t \). It is calculated using the formula \( s = r \Theta \), where \( r \) is the radius of the circle and \( \Theta \) is the angular displacement.<br /><br />### **Linear Speed (\( v \))**: <br />This is the rate at which P moves along the circle. It is calculated using the formula \( v = r \omega \), where \( r \) is the radius of the circle and \( \omega \) is the angular speed.<br /><br />## Step 2<br />Given:<br />- Radius \( r = 14 \) cm<br />- Angular speed \( \omega = \frac{\pi}{4} \) radian per sec<br />- Time \( t = 4 \) sec<br /><br />## Step 3<br />(a) **Angle Generated (\( \Theta \))**: <br />Using the formula \( \Theta = \omega t \), we get \( \Theta = \frac{\pi}{4} \times 4 = \pi \) radian.<br /><br />(b) **Distance Traveled (\( s \))**: <br />Using the formula \( s = r \Theta \), we get \( s = 14 \times \pi = 14\pi \) cm.<br /><br />(c) **Linear Speed (\( v \))**: <br />Using the formula \( v = r \omega \), we get \( v = 14 \times \frac{\pi}{4} = \frac{7\pi}{2} \) cm per sec.