Derivative - Inverse Trigonometric Functions, 5

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Prove that

Solution:

Consider the given equation above

Let

Take cosecant on both sides of the equation, we have

Take the derivative on both sides of the equation with respect to x, we have 

Next, we need to eliminate csc y cot y in the equation. We know that

but

Hence, the above equation becomes 

Finally, the equation becomes  

but

Therefore,

 

Derivative - Inverse Trigonometric Functions, 4

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Prove that

Solution:

Consider the given equation above

Let

Take secant on both sides of the equation, we have

Take the derivative on both sides of the equation with respect to x, we have 

Next, we need to eliminate sec y tan y in the equation. We know that  

but

Hence, the above equation becomes 

Finally, the equation becomes 

but

Therefore,

 

Stoichiometry Problem - Material Balance, 2

Category: Chemical Engineering Math, Algebra

"Published in Newark, California, USA"

The ground nut seeds containing 45% oil and 45% solids are fed to extractor, the cake coming out of extractor is found to contain 80% solids and 5% oil. Find the percentage recovery of oil. 

Solution:

The given word problem is about the extraction of ground nut seeds which involves the principles of Stoichiometry. The total amount of a substance in the reactants or incoming ingredients must be equal to the total amount of a substance in the final products. In short, the Law of Conservation of Mass must be followed all the time. To illustrate the problem, it is better to draw the flow diagram as follows

Photo by Math Principles in Everyday Life

Since, the amount of seeds, oil, and cake are not given in the problem, we have to assign 100 kg of seeds in the extraction process. 

Basis: 100 kg of ground nut seeds

Let x = be the amount of cake obtained
      100 - x = be the amount of oil extracted

Material Balance of Solids:

When you add the percentage composition of the ground nut seeds as well as the percentage composition of the cake, they will not be equal to 100%. The rest of the composition of the ground nut seeds and the cake maybe a water or a moisture. Since the ground nut seeds have moisture, then the oil extracted have moisture, too. We cannot use 100 - x in order to get the total amount of oil recovered because the oil extracted contains a moisture. 

Material Balance of Oil:

Therefore,