Video game company hires gamers to test a new adventure game. The number of levels completed by an average gamer is given by L = 60/(3+17e^-0.1t) where L is the number of levels completed after t hours of playing.
a) How many levels is a gamer expected to be able to complete when he is first hired? (If a gamer has not yet tested the game, he played 0 hours.)
b) How many hours would an average gamer need to play the game to complete 10 levels?
a)
L = 60/(3+17e^-0.1t)
Substitute t=0, remember that any number to the power 0=1
b)
L = 60/(3+17e^-0.1t)
substitute L=10
10=60/(3+17e^-0.1t)
3+17e^-0.1t=6
17e^-0.1t = 3
e^-0.1t=3/17
ln(e^-0.1t)=ln(3/17)
-0.1t=ln(3/17)=-1.7346
0.1t=1.7346
t=17.346 hours
But check the maths!
succ
To find the number of levels a gamer is expected to be able to complete when they are first hired (0 hours of playing), we need to substitute t = 0 into the equation.
a) Substituting t = 0 into the equation L = 60 / (3 + 17e^(-0.1t)), we get:
L = 60 / (3 + 17e^(-0.1*0))
L = 60 / (3 + 17e^0)
L = 60 / (3 + 17 * 1)
L = 60 / (3 + 17)
L = 60 / 20
L = 3
Therefore, a gamer is expected to be able to complete 3 levels when they are first hired and have not yet tested the game.
b) To find out how many hours an average gamer would need to play the game to complete 10 levels, we need to solve the equation L = 10, where L is the number of levels completed after t hours of playing.
10 = 60 / (3 + 17e^(-0.1t))
First, let's multiply both sides of the equation by (3 + 17e^(-0.1t)) to eliminate the fraction:
10 * (3 + 17e^(-0.1t)) = 60
Next, we can simplify the equation:
30 + 170e^(-0.1t) = 60
Then, rearrange the equation to isolate the exponential term:
170e^(-0.1t) = 60 - 30
170e^(-0.1t) = 30
Divide both sides of the equation by 170:
e^(-0.1t) = 30 / 170
e^(-0.1t) = 3 / 17
To solve for t, we can take the natural logarithm (ln) of both sides:
ln(e^(-0.1t)) = ln(3 / 17)
Using the property that ln(e^x) = x, we simplify the left side of the equation:
-0.1t = ln(3 / 17)
Now, divide both sides of the equation by -0.1:
t = ln(3 / 17) / -0.1
Calculating this using a calculator or math software, we find:
t ≈ -3.47 hours
Since time cannot be negative, we can discard the negative value. Therefore, an average gamer would need to play approximately 3.47 hours to complete 10 levels.