More Pyramid Problems

Category: Solid Geometry

"Published in Newark, California, USA"

What fraction of the volume of a pyramid must be cut off by a plane parallel to the base if the pyramid thus formed has a lateral area equal to one half of the lateral area of the original pyramid?

Solution:

To illustrate the problem, it is better to draw a pyramid with any base as follows

Photo by Math Principles in Everyday Life

If B₁ is parallel to B₂, then the two pyramids are similar. We can relate their lateral areas with their altitudes using the principles of ratio and proportion as follows

where S₁ and S₂ are the lateral area of the two pyramids, respectively. If S₂ = 2S₁, then the above equation becomes

Take the square root on both sides of the equation, we have

We can relate also their volumes with their altitudes using the principles of ratio and proportion as follows

but

Finally, their volume ratio is

Indeterminate Form - Zero Over Zero, 2

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Evaluate 

Solution:

Consider the given equation above

Substitute the value of x which is 0 to the above equation, we have

Since the answer is ⁰/₀, then it is considered as Indeterminate Form which is not accepted as a final answer in Mathematics. Remember this, any number, except zero divided by zero is always equal to infinity. If the Indeterminate form is either ⁰/₀ or ^∞/_∞, then we can use the L'Hopital's Rule in order to solve for these type of Indeterminate Forms. In this case, we can apply the L'Hopital's Rule for the above equation as follows

Substitute the value of x which is 0 to the above equation, we have

Therefore,

Subtended Angle Problem

Category: Trigonometry

"Published in Newark, California, USA"

A vertical pole is 35 ft. high and stands on sloping ground. It is braced by a guy rope which extends from the top of the pole to a point on the ground 25 ft. from the foot of the pole. The pole subtends an angle of 30° at the point where the guy rope reaches the ground. Determine the length of the rope.

Solution:

The given word problem is about subtended angle. Subtended angle is the angle formed by an object at a given external point. Slopping ground means that the ground is not horizontal and it is slanted or inclined. To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The above figure forms an oblique triangle. Analyze further the figure, we have

Photo by Math Principles in Everyday Life

From the given items of an oblique triangle, use Sine Law in order to solve for angle A as follows

We know that the sum of the angles of a triangle is always equal to 180°. If we know the two angles of a triangle, then we can solve for the other angle as follows

Finally, use Sine Law again in order to solve for the length of a rope as follows