Notice that k should be written as a percentage in decimal form.
B
b
Solve the equation equal to 25 and solve.
C
c
What variable depends on the individual?
A
a
After two days: 8 names
After eight days: 20 names
B
b
Number of days: About 20
Explanation: See solution.
C
c
See solution.
Practice makes perfect
a
For our exercise, the number of tasks to be learned is the amount of names you have to memorize, c= 25. The rate of learning is 20 %, which we write as k= 0.2.
f(t)= 25(1-e^(- 0.2t))
By substituting t=2 and t=8 in the function, we can approximate how many complete names you will know after 2 and 8 days, respectively.
t
25(1-e^(-0.2t))
f(t)
2
25(1-e^(-0.2( 2)))
≈ 8
8
25(1-e^(-0.2( 2)))
≈ 20
After 2 and 8 days you will know about 8 and 20 names, respectively.
b
To learn all the names, we want f(t)= 25.
25=25(1-e^(-0.2t))
By writing the left-hand and right-hand side of this equation as functions, we can graphically determine the x-coordinate where they are equal. Push Y= on the calculator and write the functions in the first two rows. If you then push GRAPH the calculator draws them in a coordinate plane.
Notice that the two functions will converge somewhere along y=25 and likely beyond x=10. Therefore, we will resize the window by pushing WINDOW and changing the settings.
It looks like the lines might intersect somewhere after t=20. Let's try to calculate this point by selecting the intersect option. Push 2nd and TRACE, then choose the list's fifth option.
This means the lines do not intersect. In fact, our function will come infinitely close to y=25 but never intersect it. Therefore, f(t) is an asymptote to y=25.
However, just because the function mathematically will never be 25 it does not mean that you will never be able to learn the 25 names. This is a limitation to the model. When f(t) is close enough to y=25 we can round up to 25 names. As previously stated, it looks like the two functions are close enough at x=20.
c
Whether or not the model applies to your own learning rate depends on how fast you can learn something new. The only variable that is personal to you would be k, the rate of learning. All other variables depend on the situation at hand.
