Area - Polar Functions, Curves

Category: Integral Calculus, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the area of a four leaf rose

Solution:

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

Photo by Math Principles in Everyday Life

In this problem, we will get the area of one leaf and then multiply it by 4 later because the four leaves are symmetrical or identical to each other. Since the limits for one leaf are not given, then we have to solve these first as follows

set r = 0, we have

Since we have the limits for one leaf of a four leaf rose, we can further label the figure as follows

Photo by Math Principles in Everyday Life

The area of the sector is given by the formula

Integrate on both sides of the equation to get the area of one leaf, we have

Since one leaf is also symmetrical, then we can rewrite the above equation as follows

Therefore, the area of a four leaf rose is

Money - Investment Problem, 3

Category: Algebra

"Published in Newark, California, USA"

Noel invested ₱ 25,000 in 2 business ventures one earning 6% and the other 8% at the end of the year. If his proceeds amounted to ₱ 1,900.00, how much did he invest in each?

Solution:

The given word problem is about money and investment problem. Let's analyze the given word problem as follows

Let x = amount of money invested in business A
      y = amount of money invested in business B
      ₱ 25,000.00 = total amount of money invested
      6% = annual interest in business A
      8% = annual interest in business B
      0.06x = annual earning in business A
      0.08y = annual earning in business B
      ₱ 1,900.00 = annual total earning

From the word statement, "Noel invested ₱ 25,000 in 2 business ventures....", then the working equation will be

                                    x + y = 25,000

From the word statement, "If his proceeds amounted to ₱ 1,900.00.....", then the working equation will be

                                    0.06x + 0.08y = 1,900

We can solve for the value of x and y using the two equations, two unknowns. Consider the first equation

                                    x + y = 25,000

                            or           y = 25,000 - x

Substitute the value of y to the second equation, we have

                                    0.06x + 0.08y = 1,900

                    0.06x + 0.08(25,000 - x) = 1,900

                         0.06x + 2,000 - 0.08x = 1,900

                                              - 0.02x = 1,900 - 2,000

                                              - 0.02x = - 100

                                                       x = 5,000

Substitute the value of x to the first equation, we have

                                     y = 25,000 - x

                                     y = 25,000 - 5,000

                                     y = 20,000

Therefore, Noel invested ₱ 5,000.00 in business A and ₱ 20,000.00 in business B.

Note: The monetary sign, ₱ means Philippine Pesos and the word problem was in 1964.

           

Sphere - Circular Section Problem

Category: Solid Geometry

"Published in Newark, California, USA"

Find the area of a section cut from a sphere of radius R by a plane distant R/2 from the center of the sphere.

Photo by Math Principles in Everyday Life

Solution:

From the given figure above, we need to get the radius of the circular section in order to get its area. Let's consider the following procedures in order to get the radius of a circular section. 

From point O which is the center of a sphere, draw a vertical line and label its end as point A. 

At the end of a circular section, label as point B and then connect to the center of a sphere at point O. OB is also the radius of a sphere. In this case, OB = R.

From point B, draw a horizontal line towards to line OA and label their intersection as point C. CB is the radius of a circular section. In this case, CB = r.

Finally, label further the above figure as follows

Photo by Math Principles in Everyday Life

To solve for r which is the radius of the circular section, use Pythagorean Theorem as follows

Therefore, the area of a circular section is