Angle - Two Intersecting Curves

Category: Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California"

Find the angle of intersection between the pair of curves:

Solution:

To illustrate the problem, it is better to sketch the graph of two curves as follows

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Next, we need to get their point of intersection by solving the two systems of equations as follows

but

The above equation becomes

Using the second equation, the value of y is 2. Therefore, their point of intersection is (2, 2). Label further the figure as follows

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The slopes of two curves can be obtained by getting their derivative with respect to x as follows

for 

then

for

then

but x = 2 from their point of intersection, therefore

The angle of intersection is given by the formula

Substitute the values of m1 and m2 to the above equation, we have

Therefore,

or

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

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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?

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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

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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