## Problem 662

Find the area A of the sector of a circle of radius 30 feet formed by the central angle radian.

Solution:-

A central angle is an angle whose vertex is at center  of a circle.

The rays of a circle angle subtend an arc on the circle. The part of the circle between the rays of the angle and the arc subtended is called a sector. We have the following theorem.

Area of a Sector

The area A of a sector of a circle of radius r formed by a central angle of radians is

A =

The  value of r in the formula is 30 feet.

The value of in the formula is radian.

Therefore,

A (area) =

= 34.615 .

## Problem 661

Find the length s of the arc of a circle of radius 85 centimeters subtended by the center angle .

Solution:-

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. We have the following theorem.

ARC LENGTH

For a circle of a radius r, a center angle of  radian subtends an arc whose length s is

s = r

Convert angle in degrees to radians.

S (arc length)= r

= 17 centimeters

53.407

## Problem 660

Find the central angle which subtends an arc of length 53 miles of a circle of radius 51 miles.

Solution:-

A center angle is a angle whose vertex is at the center. The rays of a central angle subtend (intersect) an arc on the circle. The arc length for a circle of radius r and a central angle of radians is given by s = r.

Solving this expression for given = . Substitute the values for s and r.

=

= (Substitute s = 53 and r = 51.)

1.039.

Therefore, the central angle which subtends an arc of length 53 miles of a circle of radius 51 miles is = 1.039 radians.

## Problem 659

Find the length s of the circle of radius 75 meters subtended by the center angle radian.

Solution:-

A center angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend an arc on the circle. We have following theorem.

ARC LENGTH

For a circle of a radius r, a center angle of  radian subtends an arc whose length s is

s = r

The value of r in the formula is 75 meters.

The value of in the formula is radian.

= 3 meters

## Problem 658

5.63

Solution:-

Consider a circle of radius r. A center angle of 1 revolution (360 degree ) will subtend an arc equal to the circumference of the circle. Because the circumference of the circle equals  , we see that 360 degree= .

Converting from degree to radian can be summarized as follows.

Therefore, the following is true.

= 5.63*

## Problem 657

Convert the angle in degrees to radians.

Solution:-

Consider a circle of radius r. A center angle of 1 revolution (360 degree ) will subtend an arc equal to the circumference of the circle. Because the circumference of the circle equals  , we see that 360 degree= .

Converting from degree to radian can be summarized as follows.

Now convert the given angle in degree to radian. Use the formula from above to write 1 degree in radians.

= -19 * 1 degree

## Problem 656

If a particle has a speed of r feet per second and travels a distance d ( in feet) in  time t (in second), than d = ?

Solution:-    d = r . t

## Problem 655

What is the formula for the circumference C of a circle of radius r? What is the formula for area A of a circle of radius?

Solution:-

The formula for the circumference C of a circle of radius r is C = 2r.

The formula for the area A of a circle of radius r is A =

## Problem 654

#### Find an equation for the plane y = 5x  in cylindrical coordinates.?

Solution:-

x = r cosΘ
y = r sinΘ
z = z

r sinΘ = 5rcosΘ

The plane y = 5x in cylindrical coordinates is expressed as:
tanΘ = 5

## Problem 653

Find the appropriate angle between the vectors.

u = (3, -5, 2), and (-9, 5, 1)

Solution:-

To find the angle we use the identity

=