Solving Logarithmic Equation, 2

Category: Algebra

"Published in Suisun City, California, USA"

Solve for the value of x for

Solution:

Consider the given equation above

Apply the Laws of Logarithm for the given equation above, we have

Take the inverse logarithm on both sides of the equation to the base 2, we have

Check:

Improper Integral

Category: Integral Calculus

"Published in Suisun City, California, USA"

Evaluate

Solution:

The given equation above is Improper Integral where the limits are - ∞ and + ∞. In this case, we can split the limits as follows

Consider

Since the lower limit is -∞, we can rewrite the above equation as follows

Consider

Since the upper limit is +∞, we can rewrite the above equation as follows

Therefore,

Since the given Improper Integral has a value, then the given Improper Integral is convergent. 

Word Problem - Wine Dilution

Category: Algebra

"Published in Newark, California, USA"

A Chinese wine merchant has a cask full of brandy. He draws 5 liters and fills the cask with water. Once again, he draws out another 5 liters and fills the cask with water. How many liters does the cask hold if there are now 16 liters of brandy left?

Solution:

The given word problem describes the dilution of a substance which is a brandy with water. A certain amount of brandy is drawn from the cask and replaced with the equal amount of water. Let's analyze the word problem as follows

Let x = be the amount of brandy in liters before it is drawn
      x = be the volume of the cask or container in liters

Amount of brandy at the first drawn

Amount of brandy at the second drawn

The terms inside the bracket is the amount of the brandy left after the first drawn.

Multiply both sides of the equation by x, we have

Using Quadratic Formula,

Choose the positive sign because the initial amount of brandy must be more than 16 liters

Therefore, the amount of brandy before it is drawn is 25 liters. This is also the volume or size of the cask.