Intersection - Line, Parabola

Category: Analytic Geometry, Algebra

"Published in Suisun City, California, USA"

Find the coordinates of the points of intersection for

Solution:

To illustrate the problem, let's draw the graph of the two given equations above.

For parabola, we need to reduce into standard form as follows

The center of parabola is C(0, -2) with vertex V(0, -1¾), and the ends of the latus rectum L₁(½, 0) and L₂(-½, 0). The curve of a parabola opens upward because x² is positive.

For a line, we need to express into slope-intercept form as follows

The slope of a line is 2 and the y-intercept is 1. 

From the detailed information of a parabola and a line, we can sketch the graphs as follows

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As you can see that there are two points of intersection for a parabola and a line. To get the coordinates of their points of intersection, let's solve for the systems of the given two equations as follows

Change the sign of the second equation and then proceed with the addition as follows

Equate each factor to zero and solve for the value of x

For  x - 3 = 0                                              For  x + 1 = 0            
            x = 3                                                           x = -1

Next, substitute the values of x to either one of the given two equations in order to get the value of y. In this case, we will use the equation of a line so that it is easier to get the value of y as follows

If x = 3, then y will be

If x = -1, then y will be

Therefore, their points of intersection are P₁(3, 7) and P₂(-1, -1). 

Solving Vector Problems

Category: Trigonometry

"Published in Newark, California, USA"

A pilot flew in the direction of 140° from A to B and then in the direction of 240° from B to C. A is 500 miles from B and 800 miles from C. Find the distance of the last flight and the direction from A to C. 

Solution:

The first thing that we have to do is to analyze the problem very well. The given angles are expressed as course. Typically course is measured in degrees from 0° clockwise to 360° in compass convention (0° being north, 90° being east).  
Let's draw the figure to analyze the given word problem as follows

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Complete further the labeling of the above figure as follows

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Next, use Cosine Law to solve for the distance of the last flight from A to C

Therefore, the distance of the last flight from A to C is 1,040.0154 miles.

Finally, use Sine Law to solve for the direction of the last flight from A to C

or

Therefore, the direction of the last flight from A to C is 140° + 49°14'50" = 189°14'50".

Right Circular Cylinder - Right Circular Cone, 2

Category: Solid Geometry

"Published in Newark, California, USA"

An ink bottle is in the form of a right circular cylinder with a large conical opening as shown in the figure. When it is filled with the bottom of the opening, it can just be turned upside down without any ink spilling. Prove that the depth of the cone is ³/₅ the depth of the bottle.

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Solution:

When you flipped the ink bottle, there should be no spills of ink as shown below

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The vertical section of the above figure will be like this

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As you can see in the figure that the level of the ink is exactly at the tip of the right circular cone. If it is above the tip of the right circular cone, then there will be the spillage of the ink. Let's label further the given figure as shown below

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                  Volume before Inversion = Volume after Inversion

but

The above equation becomes

Therefore