Word Problem - Number Problem

Category: Algebra

"Published in Newark, California, USA"

Find a 2-digit number whose ten's digit is 3 more than the unit's digit. The sum of the 2 digits is 7.

Solution:

Well, the given word problem is about a number problem where you will find the value of ten's and one's digits. Let's analyze the word problem as follows

Let        x = value of one's digit
       x + 3 = value of ten's digit

From the word statement, "the sum of the 2 digits is 7", the working equation for the given problem is

                                        x + x + 3 = 7

                                            2x + 3 = 7

                                                  2x = 7 - 3

                                                  2x = 4

                                                    x = 2

One's digit = x = 2
Ten's digit = x + 3 = 2 + 3 = 5

Therefore, the number is 52. The sum of the digits is 5 + 2 = 7 which is correct. 

Approximate Value - Logarithm

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the approximate value of log 97 without using a calculator or logarithm table.

Solution:

We know that the log 100 is 2. How do you get a logarithm of a number without using a calculator or logarithm table? Is that possible? Yes, we can get the approximate value of logarithm of any number (except ≤ 0) as follows

Let

then

If   x = 100
   dx = 97 - 100 = -3
     e = 2.7182818284590452353602874713527....

then dy will be

Therefore

using a calculator

from logarithm table

Finding Equation - Ellipse

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of an ellipse if the center is C(1, 3), vertex is V(1, -1), and passing through the origin. 

Solution:

To illustrate the problem, let's draw the graph as follows

Photo by Math Principles in Everyday Life

Since the major axis of an ellipse is parallel to y-axis, the equation of an ellipse in standard form is

But C (1, 3), the above equation becomes

Since one of the vertex of an ellipse is given, we can solve for the value of a as follows

Substitute the value of a to the above equation

To solve for the value of b, substitute the values of x and y which is one of the point, P(0, 0) of an ellipse as follows

Therefore, the equation of an ellipse in standard form is

You can express the equation of an ellipse in general form by expanding the above equation as follows

Multiply both sides of the equation by 16, we have