Question

A gamer decides to do some level grinding (hunting enemies to get experience points). He starts with 6000 experience points, and kills enough enemies every ten minutes to get 750 more experience points. With a time interval of one minute, a and assuming linear growth:A) Write a recursive equation for the number of experience points he has.B) Write an explicit equation for the number of experience points he has C) How many experience points will he have after 32 minutes?D) If he levels up 8000 xp, how many minutes before he levels up?

Answer 1

So first of all let's use t for the minutes he has played and E(t) for the experience points got up until t minutes. We are being told that by grinding for 10 minutes he gets 750 points and that we must assume linear growth. This means that in order to now the experience point gain per minute we just need to find:

`\frac{750\text{ xp points}}{10\text{ minutes}}=75\frac{\text{ xp}}{\text{ min}}`

We are going to use this value for making the equations.

In part A we must find a recursive equation. A recursive equation uses the value of the same equation but at an earlier time. This means something like this:

`E(t)=E(t-1)+\text{ xp gained in a minute}`

Where E(t-1) are the xp points gained up until 1 minute before minute t. Here we just need to add the amount of xp he gains in a minute that is 75. Then the recursive equation for part A is:

`E(t)=E(t-1)+75`

For part B we need to find an explicit equation. Since the xp gain follows a linear growth then this equation has the following form:

`E(t)=m\cdot t+b`

Where m is the amount of xp gained per minute and b is the base value of xp he had when he started grinding. Using the data from the question and the first calculations we made we have m=75 and b=6000 which means that the explicit equation for part B is:

`E(t)=75t+6000`

In part C we have to calculate how many xp points he'll have after 32 minutes. This basically means taking t=32 in the equation above:

`\begin{gathered} E(t)=75t+6000 \\ E(32)=75\cdot32+6000=8400 \end{gathered}`

So after grinding for 32 minutes he will have 8400 experience points.

So for part D we know that he has gained 8000 xp points and we need to find how many minutes it took him to reach that amount. In this case we do not know the value of t, we have to calculate it from the following equation:

`E(t)=8000=75\cdot t+6000`

So we have:

`\begin{gathered} 8000=75\cdot t+6000 \\ 8000-6000=75t \\ 2000=75t \\ (2000)/(75)=t \\ t=26.67 \end{gathered}`

So it took him 26.67 minutes to gain 8000 experience points.