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Sunday, January 6, 2013

Ratio, Proportion - Elongation Problem

Category: Algebra, Strength of Materials

"Published in Newark, California, USA"

The elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area. A steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it. A certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs. Compute the minimum permissible area of the member.

Solution:

To illustrate the problem, let's draw the figure as follows


Photo by Math Principles in Everyday Life

As you can see in the figure that when you applied a force F at the end of a metal with length L, there will be an elongation with length ∆L. From the first statement of the word problem, "the elongation of any metal is directly proportional to the product of the applied force F and the length L, and inversely proportional to the beam cross-sectional area," the working equation can be written as follows


and

Combining the two equations above, we have


or

where

     ∆L = elongation of a metal
       k = proportionality constant
       F = applied force
       L = length of a metal
       A = cross sectional area of a metal

From the second statement of a word problem, "a steel bar 2 square inches in cross section and 20 inches in length is elongated by 0.03 inches when a force of 10,000 lbs. was applied to it," substitute the given items to the working equation in order to get the value of k as follows






or

From the third statement of a word problem, "a certain member of the same metal whose length is 3 feet is allowed a maximum elongation of 0.5 inch when subjected to a force of 18,500 lbs," substitute the given items to the working equation in order to get the value of A as follows











Therefore, the minimum permissible area of the member is 0.1998 in2


Saturday, January 5, 2013

Indeterminate Form - Zero Raised Zero

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate 

Solution:

Consider the given equation



Substitute the value of x to the above equation



Since the answer is 00, then it is also another type of Indeterminate Form and it is not accepted as final answer in Mathematics. We know that any number raised to zero power is always equal to one except for zero that why it is also 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, we have



Substitute the value of x to the above equation



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



Substitute the value of x to the above equation



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





Substitute the value of x to the above equation





Take the inverse natural logarithm on both sides of the equation, we have



Therefore,





Friday, January 4, 2013

Solving 2 x 2 Determinants

Category: Algebra

"Published in Newark, California"

Solve the following systems of equations by determinants:

                                        x - 2y = 7

                                        3x - y = 11

Solution:

The first thing that we have to do is to write the determinants for Dx, Dy, and D from the given two linear equations. Determinant is a value associated with square matrix. Matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Consider the given equations


                                        x - 2y = 7

                                        3x - y = 11

To write the determinant of D, consider the coefficients of x and y as follows



To write the determinant of Dx, replace the coefficients of x with the coefficients of the right side of the equation as follows



To write the determinant of Dy, replace the coefficients of y with the coefficients of the right side of the equation as follows




Next, solve for the value of x as follows









Finally, solve for the value of y as follows









Note: To get the value of a 2 x 2 Matrix, principal diagonal (top left term times bottom right term) minus secondary diagonal (bottom left term times top right term).

Check: To see if you got the correct answers, substitute the values of x and y to either of the two given equations as follows

                                        x - 2y = 7

                                     3 - 2(-2) = 7

                                         3 + 4 = 7

                                               7 = 7