Partial Differentiation - Three Variables

Category: Differential Calculus

"Published in Newark, California, USA"

If

show that 

Solution:

If you will take the partial derivative of a function with respect to a certain variable, then the other variables will be treated as a constant. Consider the given equation

Take the partial derivative with respect to x, consider y as a constant as follows

Take the partial derivative with respect to y, consider x as a constant as follows

To prove that

Substitute the values of the partial derivative and the value of z to the above equation, we have

Therefore,

Work Problem - Farming

Category: Algebra

"Published in Newark, California, USA"

Joaquin can plow his farm in 4 days. He and his eldest son can finish plowing the farm in 2 days. With his youngest son helping, the three can finish the work in 1 ½ days. Alone, how long it will take the youngest son to plow the farm?

Solution:

The word problem is about a work problem. If there are at least two or more people to work in a certain job, then they will finish their work in lesser time. If there are lesser people to work in a certain job, then they will finish their work in longer time. Because of these statements, the working equation for this type of problem is rational linear equation. Let's start to analyze the word problem as follows

Joaquin can finish his work alone = 4 days
Joaquin and his eldest son = 2 days
Joaquin, his eldest son, and youngest son = 1 ½ days 
His eldest son can finish his work alone = x days
His youngest son can finish his work alone = y days

Next, write the working equation for Joaquin and his eldest son in order to get the number of days that his eldest son can plow the farm alone as follows

Multiply the both sides of the equation by their LCD, which is 4x as follows

His eldest son can plow the farm alone in 4 days also. Finally, write the working equation for Joaquin, his eldest son, and his youngest son in order to get the number of days that his youngest son can plow the farm alone as follows

Multiply the both sides of the equation by their LCD, which is 6y as follows

Therefore, his youngest son can plow the farm alone in 6 days. 

Solving Logarithmic Equation

Category: Algebra

"Published in Newark, California, USA"

Find the solution set of the equation:

Solution:

The first thing that we have to do is to transpose the constant at the right side of the equation as follows

Since the left side of the equation have the same base which is x, then we can combine and apply the laws of logarithm as follows

Take the inverse logarithm on both sides of the equation to the base x as follows

The roots of the equation are: x = 3 and x = -1.

Next, check the roots of the equation by substituting the value of x to the given equation as follows

If x = 3, then

                        x = 3 is the root of the equation

If x = -1, then

                       x = -1 is an extraneous root

Therefore, the answer is x = 3.