Angle - Elevation Problem

Category: Trigonometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

A tower of height h stands on level ground and is due north of point A and due east of point B. At A and B, the angles of elevation of the top of the tower are α and β, respectively. If the distance AB is c, show that 

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Solution:

In the given figure above, there are three right triangles involved. Let's solve for the height h in terms of α, β, and c as follows

By Pythagorean Theorem

Take the square root on both sides of the equation, we have

Therefore,

Geometric Progression - Money Problem

Category: Algebra

"Published in Suisun City, California, USA"

A very patient woman wishes to become a billionaire. She decides to follow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

Solution:

From the given word problem, let's analyze the statements as follows

1st Day = 1 cent
2nd Day = 2 cents
3rd Day = 4 cents
4th Day = 8 cents
5th Day = 16 cents
.
.
.
and so on

As you noticed that everyday she double the money that she put for her scheme. The given word problem is an application of Geometric Progression because the sequence has a common ratio which is 2. 

Let  a = 1
       r = 2

The Sum of Geometric Progression is given by the formula

where n = nth term in the sequence

Consider again the word problem that if a woman is following her scheme for 30 days, the total amount of her money will be as follows

Substitute a = 1, r = 2, and n = 30 to the above equation, we have

Therefore, in 30 days, she will earn 1,073,741,823 cents or US $ 10,737,418.23.

If she wants to become a billionaire, she must continue her scheme as follows

Take Natural Logarithm on both sides on the equation

Therefore, she will be a billionaire for 37 days!

Graphical Sketch - Ellipse, 2

Category: Analytic Geometry, Algebra

"Published in Suisun City, California, USA"

Sketch the graph for the conic section

Solution:

The given equation above represents a conic section. The general equation of any conic sections is given by the equation

To identify the type of a conic section, let's consider the following conditions, if

A = C and B = 0, then it is a Circle
B² - 4AC = 0, then it is a Parabola
B² - 4AC < 0, then it is an Ellipse
B² - 4AC > 0, then it is a Hyperbola

Now, let's identify the type of conic section for the given equation above as follows

B² - 4AC = (192)² - 4(153)(97)
               = 36864 - 59364
               = - 22500

Since B² - 4AC < 0, then the given equation is an Ellipse.

Since the equation of an Ellipse has an xy term, then the major and minor axes of an Ellipse are not parallel to x-axis and y-axis. First, let's get the angle of inclination of an Ellipse as follows

Next, get the value of Cos 2θ as follows

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Use the Half-Angle Formula to get the values of Sin θ and Cos θ as follows

Next, we need to simplify the given equation in order to eliminate the xy term. The equations that we will use are the following

and

Substitute the values of Sin θ and Cos θ to the two equations above

and

Substitute the two equations above to the given equation, we have

Divide both sides of the equation by 225, we have

Since the value of a² is at y'², then the major axis is parallel to y' axis.

To draw the x' axis, we will use the values of Sin θ and Cos θ. Count 4 units to the right from the origin since the center of an Ellipse is C(0, 0) and then 3 units upward. Connect that point with the origin and we have now x' axis.

To draw the y' axis, we will use the values of Sin θ and Cos θ. Count 4 units upward from the origin since the center of an Ellipse is C(0, 0) and then 3 units to the left. Connect that point with the origin and we have now y' axis.

In the next procedure, we will use the x' axis and y' axis to sketch the graph of an Ellipse.

If

then

If

then

The distance of the two foci from the center of an ellipse, c is calculated as follows

The distance of the four end points of latera recta from the two foci of an ellipse, p is calculated as follows

Label the center, four vertices, two foci, and the four ends of the latera recta using the x' axis and y' axis. The graph of an Ellipse will be like this

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