Archive for April, 2015

Problem 662

Find the area A of the sector of a circle of radius 30 feet formed by the central angle \frac{1}{13}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 \theta radians is

A = \frac{1}{2}r^{2}\theta

 

The  value of r in the formula is 30 feet.

The value of \theta in the formula is \frac{1}{13} radian.

Therefore,

A (area) = \frac{1}{2}.(30 feet)^{2}*\frac{1}{13}

= 34.615 feet^{2}.

 

Problem 661

Find the length s of the arc of a circle of radius 85 centimeters subtended by the center angle 36^{\circ}.

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  \theta radian subtends an arc whose length s is

s = r\theta

 

Convert angle in degrees to radians.

1^{\circ} = \frac{\pi}{180} radian

36^{\circ}= 36*= \frac{\pi}{180} radian

\approx\frac{\pi}{5} radian

 

S (arc length)= r\theta

= 85 centimeters.\frac{\pi}{5}radian

= 17\pi centimeters

\approx53.407

 

Problem 660

Find the central angle \theta 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 \theta radians is given by s = r\theta.

 

Solving this expression for \theta given \theta = \frac{s}{r}. Substitute the values for s and r.

\theta = \frac{s}{r}

= \frac{53 miles}{51miles}(Substitute s = 53 and r = 51.)

\approx 1.039.

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

 

 

Problem 659

Find the length s of the circle of radius 75 meters subtended by the center angle \frac{1}{25}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  \theta radian subtends an arc whose length s is

s = r\theta

 

The value of r in the formula is 75 meters.

The value of \theta in the formula is \frac{1}{25} radian.

S(arc length) = 75 meters*\frac{1}{25}radian

= 3 meters

Problem 658

Convert the angle in radians to degrees. Express  your answer in decimal form.

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  2\pi r, we see that 360 degree= 2\pi .

Converting from degree to radian can be summarized as follows.

1 degree = \frac{\pi }{180} radian

Therefore, the following is true.

5.63 radians = 5.63 * radian

= 5.63*\frac{180}{\pi}

\approx 322.58  ^{\circ}

 

Problem 657

Convert the angle in degrees to radians.

-19^{\circ}

 

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  2\pi r, we see that 360 degree= 2\pi .

Converting from degree to radian can be summarized as follows.

1 degree = \frac{\pi }{180} radian

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

-19^{\circ} = -19 * 1 degree

= -19 * \frac{3.1416}{180}radian

\approx -0.33 radian.

 

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 = 2\pir.

 

The formula for the area A of a circle of radius r is A = \pi r^{2}

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 u\cdot v=\left | u \right |\left | v \right |cos\Theta .

cos\Theta =\frac{(\left | u \right |\left | v \right |)}{(u\cdot v)} =\frac{-50}{\sqrt{38}\sqrt{107}}

= \Theta \approx 2.47radians \approx 141^{\circ}