More Triangle Problems

Category: Plane Geometry, Algebra

"Published in Suisun City, California, USA"

A right triangle has an area of 54 in². The product of the lengths of the three sides is 1620 in³. What are the lengths of its sides?

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

We know that the area of a right triangle is

If the product of the sides of a triangle is 1620 in³, then we can solve for the length of hypotenuse as follows

Use Pythagorean Theorem in order to solve for the other sides of a right triangle as follows

To solve for a and b, we need to use another equation which is 

or

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

Multiply both sides of the equation by a², we have

Use Quadratic Formula in order to solve for a² as follows

If we choose the positive sign,

The other side of a right triangle is

If we choose the negative sign,

The other side of a right triangle is

Therefore, the three sides of a right triangle are 9, 12, and 15. 

Binomial Theorem, 2

Category: Algebra

"Published in Newark, California, USA"

Find the first three terms of 

Solution:

Consider the given equation above

Since the given polynomial is a trinomial, then we have to do something first in order to make this as a binomial and apply the Binomial Theorem to expand the binomial. 

To make a trinomial into a binomial, group x and -2y as follows

Since x and -2y are considered as one term in a group, then we can expand using the Binomial Theorem as follows

The above equation will be more complicated and longer if you will expand (x - 2y) completely. The more important is you know the principles of Binomial Theorem especially how to group the terms and how to expand the binomial which includes the getting the exponents of each terms as well as their coefficients. Therefore, the first three terms are

or

Integration - Algebraic Substitution, 2

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

This type of integration cannot be integrated by simple integration. We have to use the technique of integration procedures. If we will use the Integration by Parts, the above equation will be more complicated because it contains radical equation. In this type of integration, we have to use the Algebraic Substitution as follows

Let

So that 

Substitute the above values to the given equation, we have

but 

Therefore, the above equation becomes

Rationalize the denominator so that the radical equation at the denominator will be eliminated as follows

Divide the numerator and denominator by x², we have

Substitute the values of limits and therefore, the final answer is

Note: If you are getting the area bounded by the curves, then the sign must be expressed in the absolute value. The area is always positive. If you interchange the position of the limits in the proper integral, then the sign of the final answer will be change.