Indeterminate Form - Infinity Raised Zero

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution: 

Consider the given equation

Substitute the value of x to the above equation, we have

Since the answer is ∞⁰, then it is also another type of Indeterminate Form and it is not accepted as a final answer in Mathematics. We know that any number raised to zero power is always equal to one except for infinity that's why it is also an Indeterminate Form. In this type of Indeterminate Form, we cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. Since the given equation is exponential equation, let's consider the following procedure

let 

Take natural logarithm on both sides of the equation

Substitute the value of x to the above equation, we have

  
Since the Indeterminate Form is 0∙∞, we have to rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is ∞/∞, then we can now use the L'Hopital's Rule as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows

Take inverse natural logarithm on both sides of the equation

Therefore, 

Right Circular Cylinder - Right Circular Cone

Category: Solid Geometry

"Published in Newark, California, USA"

The solid shown consists of a right circular cylinder of diameter d and altitude h, surmounted by a cone of diameter  of base d and altitude h/2. 

(a) Write a formula for the volume.

(b) Find the volume, given d = 4.42 in. and h = 5.17 in.

Photo by Math Principles in Everyday Life

Solution:

Consider the volume of right circular cylinder

Consider the volume of right circular cone

If h = h/2, then

The combination of the volume of right circular cylinder and right circular cone is

If r = d/2, then the equation becomes

If d = 4.42 in. and h = 5.17., then the volume is

Word Problem - Finding Unknown Fraction

Category: Algebra

"Published in Newark, California, USA"

Find the value of a fraction whose numerator is 4 less than its denominator and which when added to ½ the value of the fraction gives a sum of ³/₁₀. 

Solution:

The first thing that we have to do is to analyze the problem very well because we will interpret a word statement into a working equation. 

Let     x = be the denominator of the unknown fraction

     x - 4 = be the numerator of the unknown fraction

From the word statement that "when added to ½ the value of the fraction gives a sum of ³/₁₀", the working equation will be  

Multiply the both sides of the equation by their Least Common Denominator (LCD) which is 10x as follows

The numerator of the unknown fraction is 5 - 4 = 1.

The denominator of the unknown fraction is 5.

Therefore, the unknown fraction is ¹/₅. 

Check: