Simple Chemical Conversion

Category: Chemical Engineering Math, Differential Equations

"Published in Newark, California, USA"

For a substance C, the time rate of conversion is proportional to the square of the amount x of unconverted substance. Let k be the numerical value of the constant of proportionality and let the amount of unconverted substance be x₀ at time t = 0. Determine x for all t ≥ 0.

Solution:

The given word problem is another good application of first order, first linear equation where a simple chemical conversion of a substance is involved. Let's analyze the given problem as follows

for                           C ────────────> Product

initial                       x₀                                     0
at time t                 x₀ - x                                  x

From the word problem, "the time rate of conversion is proportional to the square of the amount x of unconverted substance", the working equation will be

The above equation is negative because the value of x is decreasing as time is increasing. Using the method of separation of variables, we have

Integrate on both sides of the equation

To solve for the value of C, substitute x = x₀ and t = 0 to the above equation, we have

Therefore, the amount of x at t ≥ 0 will be

Proving Trigonometric Identities, 2

Category: Trigonometry

"Published in Newark, California, USA"

Prove that

Solution:

In proving the trigonometric identities, we have to simplify the hardest or more complicated part of the equation. In this case, we have to simplify the left side of the given equation. Let's consider the following procedure.

but

The above equation becomes

but

The above equation becomes

Therefore,

Area Bounded - Two Curves

Category: Integral Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the area bounded by the given curves

Solution: 

To illustrate the problem, it is better to draw or sketch the graph of two given equations above by using the principles of Analytic Geometry as follows

Photo by Math Principles in Everyday Life

Since the limits or their points of intersection are not given, then we have to solve their points of intersection using the two equations, two unknowns as follows

Subtract the second equation from the first equation, we have

Substitute the value of y to either of the two given equations in order to get the value of x as follows

Equate each factor to zero and solve for the value of x. The values of x are 3 and -1.

Their points of intersection are (-1, -4) and (3, -4).

Rewrite the two given equations as y = f(x), label further the figure, and use the vertical strip as follows

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The area bounded by the two curves is given by the formula

Substitute the values of the limits and the two functions to the above equation, we have