The "learning curve" describes the rate at which a person learns certain tasks. If a person sets a goal of typing N words per minute (wpm), the length of time t (in days) to achieve this goal is given by the following formula. t=62.5ln(1-N/80)
(a) According to this formula, what is the maximum number of words per minute?
(b) Solve for N.
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ln(x) is only defined for x>0, so 1-N/80 > 0
That is, N<80
The problem here is that 1-N/80 < 1, so ln(1-N/80) is negative. But, t must be positive.
Typo?
To find the maximum number of words per minute (WPM) according to the given formula, we need to understand that the natural logarithm function (denoted as ln) has a domain restriction. In this case, the argument of the ln function, (1 - N/80), should be a positive number.
(a) To determine the maximum WPM value, we need to find the value of N that makes the argument of the ln function equal to zero. So, we set (1 - N/80) as zero and solve for N:
1 - N/80 = 0
N/80 = 1
N = 80
Therefore, the maximum number of words per minute is 80 WPM.
(b) To solve for N, we need to rearrange the equation:
t = 62.5 * ln(1 - N/80)
First, divide both sides by 62.5:
ln(1 - N/80) = t/62.5
Next, apply the exponential function (e^x) to both sides of the equation to cancel out the ln function:
e^(ln(1 - N/80)) = e^(t/62.5)
1 - N/80 = e^(t/62.5)
Now, isolate N by subtracting 1 from both sides:
-N/80 = e^(t/62.5) - 1
Multiply both sides by -80 to get rid of the fraction:
N = -80 * (e^(t/62.5) - 1)
So, to find N, substitute the given value of t into the equation.