Graphical Sketch - Parabola

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the vertex, focus, and end points of the latus rectum for parabola with an equation 

and sketch the graph.

Solution:

The first thing that we have to do is to write the equation of a parabola in standard form as follows

Transpose -3y and 10 to the right side of the equation

Complete the square at the left side of the equation by adding 1 on both sides of the equation

Therefore, the vertex of a parabola is V(1, 3). 

Since the second degree in the equation is x, the parabola is either concave upward or downward. 

Next, let's solve for the coordinates of the focus and the ends points of the latus rectum. Since 4a = 3 is positive, then the parabola is concave upward. a is the distance between the focus and the vertex of a parabola. Solve for the value of a as follows

                                       4a = 3
                                    
                                         a = ¾

To get the coordinates of the focus, count ¾ units upward from the vertex. Therefore, the coordinates of the focus is F(1, 3¾). 

To get the coordinates of the first end point of the latus rectum, count 2a = 2(¾) = 1½ units to the right from focus. The first end point of the latus rectum is B(2½, 3¾). 

To get the coordinates of the other end point of the latus rectum, count 2a = 2(¾) = 1½ units to the left from focus. The other end point of the latus rectum is A(-½, 3¾).

Finally, we have the coordinates of the vertex, focus, and the end points of the latus rectum. We can sketch the graph of the parabola as follows

Photo by Math Principles in Everyday Life

More Right Circular Cylinder Problem

Category: Solid Geometry, Trigonometry

"Published in Newark, California, USA"

At what angle with the horizontal must the base of a right circular cylinder be tilted to make it just topple over if its diameter is 12 ft. and its altitude is 16 ft.? (The center of gravity is at the middle point C of the axis of the cylinder. The base must be tilted so that the line AC becomes vertical.)

Photo by Math Principles in Everyday Life

Solution:

By looking at the figure, we need to analyze the small triangle because we want to get the angle of tilt for right circular cylinder. Consider the figure below

Since the triangle is a right triangle, we can use the basic trigonometric function to solve for angle θ and ϕ as follows

Therefore, we can get the angle of tilt as follows

                                     θ + ϕ = 90º 

                           53.130º + ϕ = 90º

                                           ϕ = 90º - 53.130º

                                           ϕ = 36.870º

World Population Projections

Category: Algebra

"Published in Suisun City, California, USA"

The population of the world in 1995 was 5.7 billion, and the estimated relative growth rate is 2% per year. If the population continues to grow at this rate, when will it reach 57 billion?

Solution:

The word problem is about the exponential growth for a population. The exponential growth is given by the formula

                                      y = x e^rt

where

     y = population at time t
     x = initial size of population
     r = relative rate of growth (expressed as a proportion of the population)
     t = time of growth

Now, refer to the given problem that the initial population is 5.7 billion, the growth rate is 2% per year, and the final population is 57 billion. Substitute the given values to the above equation to solve for time t, we have

                                       y = x e^rt

                                     57 = 5.7 e^0.02t

                                     10 = e^0.02t

                                 ln 10 = ln e^0.02t

                                 0.02t = ln 10

                                       t = 115.129 years

Therefore, the population will reach 57 billion in approximately 115 years, that is, in the year 1995 + 115 = 2110.