Rate, Distance, Time - Problem

Category: Algebra

"Published in Newark, California, USA"

A bus bound for Baguio left Manila at exactly 5:00 am traveling at a uniform speed. Thirty minutes later, a car left Manila in pursuit of the bus. Traveling also at a uniform rate, the car overtook the bus 100 miles from the city. Had it increased its speed by 10 miles per hour, it would have overtaken the bus only 50 miles from Manila. Find the rate at which the bus was traveling.

Solution:

Well, this is a complicated problem solving about the rate, distance, and time problem. We have to analyze the given word problem as follows

Let  x = be the speed of a bus
      y = be the speed of a car
       t = time

Let's draw a simple figure to illustrate the problem

Photo by Math Principles in Everyday Life

As you noticed that we subtract 30 minutes or ½ hour for the travel time of a car because there's a 30 minute gap between a bus and a car. Since a bus left Manila first, then we have to subtract ½ hour for the travel time of a car. 

                                         Distance = speed x time

From the word statement "the car overtook the bus 100 miles from the city", the working equation for the said statement is

                              Distance of a Bus = Distance of a Car

The total time traveled by a car at uniform speed is 

If the speed of a car is increased by 10 miles per hour, then the total time will be

If you drive faster, you will arrive to your destination earlier. In this case, you save time for commuting or driving if you drive faster and you will have an excess time as well. Excess time is calculated as follows

Excess Time = Total Time at Uniform Speed - Total Time at Increased Speed

From the word statement "Had it increased its speed by 10 miles per hour, it would have overtaken the bus only 50 miles from Manila", then the working equation will be

                                           Distance = Rate x Time

If you choose a positive sign,

Total time traveled by a car

Therefore, the speed of a bus is

If you choose a negative sign,

Total time traveled by a car

Therefore, the speed of a bus is 

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

Photo by Math Principles in Everyday Life

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).