Archive for the ‘Geometry’ Category

Problem 963

Find the area of the segment of a circle whose radius is 3 feet, formed by  a central angle of 65°.

circle

 

Solution:-

 

Start by finding the area of the sector formed by a central angle 65°. The area of the sector by multiplying the area of the circle by the ratio of the central angle to 360°.

Area of Sector = \pi r^{2}\frac{Cental Angle}{360}

=\pi 3^{2}\frac{65}{360}

\approx 5.1051

The area of the sector is 5.1051 square feet.

Next find the area of the triangle. We know the lengths of two sides of the triangle, and the include angle, so use the formula

K = \frac{1}{2} a*b*sin\gamma

= \frac{1}{2} * 3 * 3 sin 65°

\approx 4.0784

The area of the triangle is 4.0784 square feet.

Subtract the area of the triangle from the area of the sector to find the area of the segment.

Area of segment  = 5.1051 – 4.0784

\approx 1.03

The area of the segment is 1.03 square feet.

 

 

Problem 806

Find all horizontal and vertical asymptotes.

f(x)=\frac{3x^{2}-16x+15}{x^{2}-2x-15}

 

Solution:-

 

The horizontal asymptote(s) is/are  y = 3.

The vertical asymptote(s) is/are = x = -3.

 

Problem 734

Show the following series diverges?

3 + 5 + 7 +9+……

 

Solution:-

 

Look at the value of the common ratio r  . If \left | r \right |<1 , then the series has a limit and will be convergent. If\left | r \right |>1 , then the series has no limit and will be

divergent.

 

Problem 716

Write the expression in a + bi form  (\sqrt[5]{4}(cos18 +isin18))^{5}

 

Solution:-

 

(\sqrt[5]{4}(cos18 +isin18))^{5}

=(\sqrt[5]{4})^{5}(cos(5*18)+i sin(5*18))

=4(cos 90+ isin90)

= 4(0+i(1))

=4i

 

Problem 715

Find   z*w . Leave your answer in simplified polar form. Express all angles in degrees rounded to the nearest tenth.

 

Solution:-

 

z * w =2(cos140+i sin 140) * 7(cos65+ isin 65 )

=(2*7)(cos(140+65)+isin(140+65))

= 14(cos205 + isin205)

 

Problem 712

A fire truck is en route to an address that is six blocks east and seven blocks south of the fire station. Using the fire station as the pole and the east direction as the polar axis, express the fire truck’s destination in polar coordinates. Round the coordinates to the nearest unit. Express θ in degrees.

 

Solution:-

 

(9, -49°)

Problem 711

Identify the conjugate of the complex number  10 + i

 

Solution:-

 

The conjugate of the complex number a + bi is the complex number a – bi  .

10 – i

Problem 706

What is the length of the minor axis of the following graph?

 

2

 

Solution:-

 

This is an ellipse with a center at (0, 0), vertices at (-5, 0) and (5, 0), co-vertices at (0, 3) and (0, -3), the major axis has a length of 10 and the minor axis has a length of 6, the foci are (-4, 0) and (4, 0).

 

Problem 705

What is the length of the major axis of the following graph?

 

1

 

Solution:-

 

This is an ellipse with a center at (0, 0), vertices at (0, -6) and (0, 6), co-vertices at (-3, 0) and (3, 0), the major axis has a length of 12 and the minor axis has a length of 6, the foci are (0,-\sqrt{27}) and (0,   \sqrt{27} ).

 

 

Problem 693

Change the logarithmic statement  log 4x = 2  into an equivalent exponential statement.

 

Solution:-

 

bx = a ↔ log ba = x

42 = x