Graph of the System of Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the solution of the system of inequalities and find the coordinates of all vertices for

a. x + y ≤ 4
          y ≥ x

b. 2x + 3y > 12
     3x - y < 21

Solution:

a. For x + y ≤ 4, we need to rewrite the given equation into slope-intercept form as follows

Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 4 which is correct and that point is included in the solution.  

For y ≥ x, the given equation is already written in slope-intercept form. If x = 0 and y = 0, then the given equation reduces to 0 ≥ 0 which is correct and that point is included in the solution. Therefore, the graph of a pair of linear inequalities is

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The vertex of the graph is (2, 2) which is the intersection of two lines. It is also included in the solution.

b. For 2x + 3y > 12, we need to rewrite the given equation into slope-intercept form as follows

Since the sign of inequality is greater than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 > 12 which is not correct and that point is not included in the solution.  

For 3x - y < 21, we need to rewrite the given equation into slope-intercept form as follows

If you divide both sides of the equation by a negative number, then the sign of inequality will be reversed. Since the sign of inequality is less than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 < 21 which is correct and that point is  included in the solution. Therefore, the graph of a pair of linear inequalities is

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The vertex of the graph which is the intersection of two lines  is not included in the solution.

Graphs of Quadratic Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the following inequalities:

a. y > x² + 1
b. x² + y² ≥ 4

Solution:

a. For y > x² + 1, the given equation is a parabola that concave upward whose vertex is V(0, -1). Since the sign of inequality is greater than, then all points along the curve are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 > 1 which is not correct and that point is not included in the solution. Therefore, the graph of the given equation is

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b. For x² + y² ≥ 4, the given equation is a circle whose center is C(0, 0) and its radius is r = 2. Since the sign of inequality is greater than or equal to, then all points along the curve are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≥ 4 which is not correct and that point is not included in the solution. Therefore, the graph of the given equation is

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Graphs of Linear Inequalities

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Graph the following inequalities:

a. y > -3
b. x ≤ 2
c. 3x + 4y + 12 > 0
d. 2x - y ≤ 8

Solution:

a. For y > -3, the given line is a horizontal line. Since the sign of inequality is greater than, then all points along the line are not included in the solution. However, all points above the given line are included in the solution. Here's the graph of the given line

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b. For x ≤ 2, the given line is a vertical line. Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. However, all points at the left of the given line are included in the solution. Here's the graph of the given line

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c. For 3x + 4y + 12 > 0, we need to rewrite the given equation into slope-intercept form as follows
 

 

 

Since the sign of inequality is greater than, then all points along the line are not included in the solution. If x = 0 and y = 0, then the given equation reduces to 12 > 0 which is correct and that point is included in the solution. Therefore, the graph of the given equation is
 

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d. For 2x - y ≤ 8, we need to rewrite the given equation into slope-intercept form as follows

If you divide both sides of the equation by a negative number, then the sign of inequality will be reversed. Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 8 which is correct and that point is included in the solution. Therefore, the graph of the given equation is

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Projectile Motion Problems

Category: Mechanics, Physics

"Published in Newark, California, USA"

An object falls from a plane flying horizontally at an altitude of 40,000 ft. at 500 mi/hr. How long will it take to hit the ground? What is the horizontal distance traveled of an object?

Solution:

The given problem is about projectile motion in which an object is thrown from its maximum height with an initial velocity. The projectile curve is a half-parabola as shown in the figure below

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First, let's consider the y-component of an object in order to calculate the time required to hit the ground. From the distance formula
 

Since an object is thrown in a downward position, then the values of s_y and a_y are negative. a_y is also equal to g = 32.174 ft/sec²  which is an acceleration due to gravity. V_0y is zero because the direction of initial velocity of an object is horizontal. Therefore, the time required of an object to hit the ground is
 

          

 

 

 

 

Next, let's consider the x-component of an object in order to calculate the horizontal distance traveled of an object. From the distance formula

Since the direction of v_0x is horizontal and constant, then a_x is equal to zero. Therefore, the horizontal distance traveled of an object is

                      or

Rate, Distance, and Time Problems, 8

Category: Physics, Mechanics

"Published in Newark, California, USA"

A runner A can run the mile race in 4.25 min. Another runner B requires 4.55 min to run this distance. If they start out together and maintain their normal speeds, how far apart will they be at the finish of the race?

Solution:

The given problem is about rate, distance, and time problem in which the two runners run in the same mile race. From the given problem also, runner A is faster than runner B because runner A has a lesser time to complete the mile race than runner B.

The speed of runner A is

The speed of runner B is

If the two runners will start together in a mile race, then runner A will finish the race. Therefore, their distance between the two runners at the end of the race is

For runner B, we use the time for runner A because runner A will finish the race.

                        or