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.       
            

Solving Exact Equations

Category: Differential Equations

"Published in Newark, California, USA"

Solve for the solution for

Solution:

The first thing that we have to do is to check the above equation if it is exact equation or not as follows

Let 

then 

Let 

then

Since

then the given equation is Exact Equation. The solution for the above equation is F = C. Consider the given equation

Let 

and

Integrate the partial derivative of the first equation above with respect to x, we have

Since we are integrating the partial derivatives, then another unknown function, T(y) must be added.  If 

then

To solve for T'(y), equate

Since the arbitrary constant is already included in F = C, then we don't have to add the arbitrary constant in the above equation. Therefore,

Therefore, the general solution is