Finding Equation - Plane

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a plane that contains the points (1, -2, 4), (4, 1, 7), and (-1, 5, 1).

Solution:

The first thing that we have to do is to write the equation of a plane that contains the first point as follows

Next, substitute the values of x, y, and z from the other two points to the above equation, we have

for (4, 1, 7), then the above equation becomes

or

for (-1, 5, 1), then the above equation becomes

If you add the two equations above, then we can solve for the value of A in terms of B as follows

Substitute the value of A to either of the two equations to solve for the value of C in terms of B as follows

Therefore, the equation of a plane is

    

Divide both sides of the equation by B and then simplify, we have

Photo by Math Principles in Everyday Life

Four Tangent Circles

Category: Plane Geometry, Algebra

"Published in Newark, California, USA"

Each of the four circles shown in the figure is tangent to the other three. If the radius of each of the smaller circles is x, find the area of the largest circle. If x = 2.71, what is the area of the largest circle?

Photo by Math Principles in Everyday Life

Solution:

When you connect the centers of the three small circles, the figure is an equilateral triangle. The center of an equilateral triangle will be the radius of a big circle as shown below

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Consider the equilateral triangle, draw the perpendicular lines from each vertex, and label further as follows

Photo by Math Principles in Everyday Life

In a 30° - 60° - 90° triangle, the shortest side is half of the hypotenuse which is ½ y. The other side of a triangle is equal to the shortest side times the square root of three. Therefore, we can solve for the value of y in terms of x as follows

or

The radius of a big circle is

Therefore, the area of a big circle is

If x = 2.71, then the area of a big circle is

Word Problem - Partition, Distribution

Category: Algebra

"Published in Newark, California, USA"

Alberto would like to distribute a bagful of marbles equally among his friends. He figures out that if he were to give them 7 marbles each, he will need 4 marbles more but if he were to give them 6, he will have 3 marbles left. How many marbles does Alberto have?

Solution:

For this type of word problem, we will use the principles of linear equation where the partition or distribution of items is involved. Let's analyze the given word problem as follows

Let x be the number of marbles
      y be the number of friends to distribute the marbles

If the statement says "if he were to give them 7 marbles each, he will need 4 marbles more", then the working equation will be

or

If the statement says "if he were to give them 6, he will have 3 marbles left", then the working equation will be

or

Equate x = x, we have

Therefore,

Substitute the value of y to either one of the two equations, we have

Therefore, Alberto has 294 marbles that he distributed to his 46 friends.