Circle - Area Derivation

Category: Integral Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

If the equation of a circle is x² + y² = r², prove that the area of a circle is A = πr².

Solution:

To illustrate the problem, let's draw the graph of a circle as follows

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If the equation of a circle is

We can rewrite the above equation as a function of x as follows

To get the area of a circle, consider only a quadrant as follows

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The area of a curve bounded by the function, x-axis, and y-axis is given by the formula

Therefore, the area of a circle is

Rate, Distance, Time - Problem, 2

Category: Algebra

"Published in Newark, California, USA"

Jose left for San Fernando, La Union at exactly 12 pm from Quezon City, driving his own car at a constant speed. One hour later, his kid brother drove out after him. Julio, the brother, started out at 40 miles per hour, increasing his speed by 20 miles per hour every half hour until he overtook Jose at exactly 3 pm. At what rate was Jose driving? How far did Julio catch up with his brother?

Solution:

The given word problem is about rate, distance, and time problem but the principles of arithmetic progression is involved. To illustrate, the problem, let's draw a simple figure as follows

Photo by Math Principles in Everyday Life

To get the speed of Julio at 3 pm, let's use the formula to get the last term of arithmetic progression as follows

The distance traveled by Julio to catch up Jose is 

  
The constant speed of Jose is 

Circle - Circumference Derivation

Category: Integral Calculus, Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

If the equation of a circle is x² + y² = r², prove that the circumference of a circle is C = 2πr.

Solution:

To illustrate the problem, let's draw the graph of a circle as follows

Photo by Math Principles in Everyday Life

If the equation of a circle is

We can rewrite the above equation as a function of x as follows

Take the derivative of the above equation with respect to x as follows

To get the length of a curve or circumference of a circle, consider only a quadrant as follows

Photo by Math Principles in Everyday Life

The length of a curve is given by the formula

Substitute the value of ^dy/_dx to the above equation, we have

Therefore, the circumference of a circle is