Graphical Sketch - Plane

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Sketch the graph for 

Also, find the intercepts and the distance of the plane from the origin.

Solution:

The given equation above represents a plane in the x, y, and z axes. To sketch the graph of a plane, let's consider the following procedure as follows

For 

  
Set x = 0 and the equation becomes 

or

If y = 0, then z = 2 and if z = 0, then y = 4.

Set y = 0 and the equation becomes

or

If x = 0, then z = 2 and if z = 0, then x = 6.

Set z = 0 and the equation becomes

If x = 0, then y = 4 and if y = 0, then x = 6. 

Therefore the intercepts are:

                       x-intercept = 6
                       y-intercept = 4
                       z-intercept = 2

From the values of intercepts, we can plot the points in the x, y, and z axes and sketch the graph by connecting all the points as follows

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The distance from a point to a plane is given by the formula

where the sign of the radical is the opposite sign of D.

In this case if the equation of a plane is 

and a point is the origin, therefore the distance of a plane to the origin is 

The negative sign means that the plane is above the given point or origin.

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

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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