SOLUTION: Suppose that a student has 100 vocabulary words to learn. If a student learns 20 words in 30 minutes, the function L(t) = 100(1 − e −0.0074t ) models the number of words, L,

Question 1167397: Suppose that a student has 100 vocabulary words to learn. If a student learns 20 words in 30 minutes, the function
L(t) = 100(1 − e −0.0074t ) models the number of words, L,
that the student will learn after t minutes.
(a) After how long will the student learn 20 words?
(b) After how long will the student learn 70 words?

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
import numpy as np
# Define the function parameter
k = -0.0074
# Define the function to solve for time t
# L(t) = 100 * (1 - e^(k*t))
# t = ln(1 - L/100) / k
def calculate_time(L_target):
# Check for impossible scenario (L >= 100)
if L_target >= 100:
return np.inf

# Calculate the fraction part
fraction = 1 - L_target / 100

# Check for invalid logarithm input (should not happen if L < 100 and L >= 0)
if fraction <= 0:
return np.nan # Not a number

# Calculate t
t = np.log(fraction) / k
return t
# (a) After how long will the student learn 20 words?
L_a = 20
t_a = calculate_time(L_a)
# (b) After how long will the student learn 70 words?
L_b = 70
t_b = calculate_time(L_b)
# Print results rounded to two decimal places
print(f"Time to learn 20 words (t_a): {t_a:.2f} minutes")
print(f"Time to learn 70 words (t_b): {t_b:.2f} minutes")
```
```text?code_stdout&code_event_index=2
Time to learn 20 words (t_a): 30.15 minutes
Time to learn 70 words (t_b): 162.70 minutes
```
The problem requires solving the learning function $L(t) = 100(1 - e^{-0.0074t})$ for the time $t$.
The general formula to find the time $t$ for a given number of words $L$ is:
$$t = \frac{\ln\left(1 - \frac{L}{100}\right)}{-0.0074}$$
-----
## (a) After how long will the student learn 20 words?
We set $L = 20$ and solve for $t$:
$$t = \frac{\ln\left(1 - \frac{20}{100}\right)}{-0.0074} = \frac{\ln(0.8)}{-0.0074}$$
$$t \approx \frac{-0.22314}{-0.0074} \approx 30.1546$$
The student will learn 20 words after approximately **30.15 minutes**.
*(Note: The initial information that the student learns 20 words in 30 minutes is an approximation, and the function provides a more precise time of $30.15$ minutes.)*
-----
## (b) After how long will the student learn 70 words?
We set $L = 70$ and solve for $t$:
$$t = \frac{\ln\left(1 - \frac{70}{100}\right)}{-0.0074} = \frac{\ln(0.3)}{-0.0074}$$
$$t \approx \frac{-1.20397}{-0.0074} \approx 162.70$$
The student will learn 70 words after approximately **162.70 minutes**.
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