More Right Circular Cylinder Problem, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A right cylindrical solid of altitude 6 in. has the cross section shown in the shaded portion of the figure. BEDG is a circle whose radius is OG. AFCG is a circle which is tangent to the larger circle at G. If AB = CD = 5 in. and EF = 9 in., find the volume of the cylinder.

Photo by Math Principles in Everyday Life

Solution:

The cross section of a right circular cylinder consists of two tangent circles with their common point at Point G. Let's further analyze and label the cross section of a right circular cylinder as follows

Photo by Math Principles in Everyday Life

We noticed that line segments AC and FG are the chords of a small circle that intersect at point O, which is also the center of a big circle. From Plane Geometry, we know that the product of two divided chords is equal to the product of other two divided chords. We can solve for the radius of a big circle as follows

The line segment OF is calculated as follows

The radius of a small circle is calculated as follows

The area of the shaded portion of the cross section of a right circular cylinder is calculated as follows

             Area of a base = Area of Big Circle - Area of Small Circle

Therefore, the volume of a right circular cylinder is

or

Polynomials - Nested Form

Category: Algebra

"Published in Newark, California, USA"

Rewrite the given polynomial into nested form 

If x = 1, find the value of the function.

Solution:

It is possible to eliminate the exponential powers of the given polynomial by rewriting it into nested form. Nested means grouped then factored. Let's consider the given equation above

Arrange the given polynomial according to their descending power if they are not yet arranged. If any of the exponential term is missing, you must include it by adding zero times the missing exponential term. Since the given polynomial is already arranged according to their descending power, then we can proceed with the grouping and factoring as follows

Therefore, in nested form is

If x = 1, then the value of the function is

Therefore, 

Check:

More Integration Procedures, 5

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

The first thing that we have to do is to find the differential of the given equation above. If

then

Hence, the above equation becomes

but

Substitute the value of sec² x to the above equation, we have

Therefore,