Square Pyramid Problems

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

A vessel is in the form of an inverted regular square pyramid of altitude 9.87 in. and a base edge 6.27 in. The depth of the water it contains is 6 in. How much will the surface rise when 1 pint of water is added? (One gallon = 231 cubic inches) Find the wetted surface when the depth of the water is 9.23 in. 

Solution:

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

Photo by Math Principles in Everyday Life

The volume of the empty vessel is calculated as follows

When the depth of the water is 6 in, the volume is calculated by ratio and proportion as follows

When 1 pint of water is added, the volume in cubic inches is

Therefore, after the addition of 1 pint of water, the height of water can be calculated by ratio and proportion as follows

The difference of the height of water after the addition of 1 pint of water is

When the depth of the water is 9.23 in, the length of a square base is calculated by ratio and proportion as follows

By Pythagorean Theorem, the slant height will be equal to

Photo by Math Principles in Everyday Life

Therefore, the wetted surface of a vessel is 

Angle - Elevation Problem

Category: Trigonometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

A tower of height h stands on level ground and is due north of point A and due east of point B. At A and B, the angles of elevation of the top of the tower are α and β, respectively. If the distance AB is c, show that 

Photo by Math Principles in Everyday Life

Solution:

In the given figure above, there are three right triangles involved. Let's solve for the height h in terms of α, β, and c as follows

By Pythagorean Theorem

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

Therefore,

Geometric Progression - Money Problem

Category: Algebra

"Published in Suisun City, California, USA"

A very patient woman wishes to become a billionaire. She decides to follow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

Solution:

From the given word problem, let's analyze the statements as follows

1st Day = 1 cent
2nd Day = 2 cents
3rd Day = 4 cents
4th Day = 8 cents
5th Day = 16 cents
.
.
.
and so on

As you noticed that everyday she double the money that she put for her scheme. The given word problem is an application of Geometric Progression because the sequence has a common ratio which is 2. 

Let  a = 1
       r = 2

The Sum of Geometric Progression is given by the formula

where n = nth term in the sequence

Consider again the word problem that if a woman is following her scheme for 30 days, the total amount of her money will be as follows

Substitute a = 1, r = 2, and n = 30 to the above equation, we have

Therefore, in 30 days, she will earn 1,073,741,823 cents or US $ 10,737,418.23.

If she wants to become a billionaire, she must continue her scheme as follows

Take Natural Logarithm on both sides on the equation

Therefore, she will be a billionaire for 37 days!