Inverse Variation Analysis: Application to Speed, Time, and Distance

This assignment analyzes the concept of inverse variation using the rational function y = 1/x and a real-world problem involving the speed of a bicycle traveling from home to college. The student explores the relationship between time and speed, demonstrating how an increase in time leads to a decrease in speed, and vice-versa. Data is presented in a table and visualized using a scatterplot in Microsoft Excel. The analysis includes algebraic representations using the concept of limit functions to show the inverse relationship. The assignment explains how the student can calculate the required speed based on the time available, and highlights the applicability of inverse variation in this scenario. The document concludes that the model is a good fit for the data, emphasizing the inverse relationship between speed and time, which is analogous to the rational function y=1/x. The student also uses Desmos to graph the rational function and explain the concept of inverse variation.

Assignment
The graph on the rational function y = 1/x using Desmos is shown below:
The rational function y = 1/x follows inverse variation so we have to think of a real life problem that
involves this kind of variation.
One real life problem involving the use of inverse variation is the speed required to travel from one
place to another in a given amount of time.
Let’s say a student need to drive a bicycle from his home to college, which is about 5 miles. Now he has
to calculate the speed of his bicycle at which he should drive in order to get college on time given the
amount of time he has on a particular day.
This data will be the basis of our work to show the real life problem involving inverse variation.
The data will be analyzed in Microsoft Excel to graph a scatterplot.
The data is shown in the table below:

Time (in minutes) Speed (in miles/hour)
5 60
6 50
7 42.85714
8 37.5
9 33.33333
10 30
11 27.27273
12 25
13 23.07692
14 21.42857
15 20
16 18.75
17 17.64706
18 16.66667
19 15.78947
20 15
The scatterplot corresponding to this data is obtained in Microsoft Excel using Megastat which is shown
below:
4 6 8 10 12 14 16 18 20 22
0
10
20
30
40
50
60
70
Real Life problem Scatterplot
Time (in mins)
Speed (in miles/hour)

(a) As we can see from the graph of the rational function y = 1/x that is plotted with the use of Desmos
that when one quantity increases, the other decreases or we can say the two quantities are inversely
related to each other. This kind of variation is known as Inverse variation.
One value of the rational function y=1/x over time say the variable y tends to decrease over time while
the other variable x tends to increase. As it is evident from the graph that the value of the variable
quantity y approaches zero when the value of the variable quantity x tends towards infinity.
Algebraically, we can use the concept of limit function to establish this fact i.e.
lim
x→ ∞
y
= lim
x→ ∞
(1/x ¿) ¿
= 0
In our case the data also follows the same trend which can be seen from the scatterplot i.e. when the
time required reaching the college increases, the speed of the bicycle reduces in an inverse manner. The
more time student has, the slower he will go i.e. the speed will decrease.
When he has only 10 minutes to reach the college he will go at a speed of 30 miles per hour and when
he has 20 minutes, then he will go at a speed of 15 miles per hour. So our data also follows the same
trend.

(b) As we approach zero from the right, we can see this in the graph of the rational function that the
value of the variable quantity y goes on increasing to a very high magnitude i.e. infinity. This fact can be
established with the help of the concept of limit function i.e.
lim
x→ 0+¿ y ¿
¿
= lim
x→ 0+¿(1/ x)¿
¿
=+ ∞
Similarly, we see that this kind of trend is also shown in our data i.e. as the time decreases, the speed of
the bicycle increases. So hypothetically, when the time approaches zero, the speed of the bicycle tends
towards infinity which is the same trend as seen in the graph of the rational function.
The equation for figuring out how fast the student should travel from the amount of time he has is given
by:
speed= distance
time
or we can write in notation form as:
v= d
t
where
v denotes speed in miles per hour
d denotes distance travelled
t denotes the time required
So once again using the concept of limit function in our equation we have:
lim
t →0
v
= lim
t →0
( d
t )
= +∞
Thus we have established the fact algebraically that the data also follows the same trend as of the
rational equation y=1/x.

(c) Yes, this model is a good fit for my data as we have seen from the above observation that both the
graph of the rational function y=1/x and the scatterplot follow the same trend of inverse variation which
is used to predict the speed and time required for a particular day.
The equation governing the scatterplot of the data is:
speed= distance
time
which tells us that the speed and time are inversely related to each other which is similar to the rational
function:
y= 1
x
In our case we can write:
v=5
t
where the distance travelled from home to college is 5 miles.
So the concept of inverse variation is used in the real life problem involving speed and time and one can
use it to determine the speed at which one could travel for a certain amount of time and vice-versa.