UNL has decided to gradually transition from one online gradebook system to another. The percentage of professors still using the old system years after Fall 2015 is given by B(t)=100-18t. Find domain


To find the domain of the function \( B(t) = 100 - 18t \), we need to determine the range of values for which the function is defined, based on the context of the problem.

### Analysis of the Function

1. **Understanding the Context:**
   - \( B(t) \) represents the percentage of professors still using the old system.
   - \( t \) represents the number of years after Fall 2015.

2. **Mathematical Constraints:**
   - The percentage of professors still using the old system, \( B(t) \), must be between 0% and 100%. Therefore:
     \[
     0 \leq B(t) \leq 100
     \]
   - Substituting the function \( B(t) = 100 - 18t \) into the inequality:
     \[
     0 \leq 100 - 18t \leq 100
     \]

3. **Finding the Valid Range for \( t \):**

   - First, consider the lower bound \( 0 \leq 100 - 18t \):
     \[
     0 \leq 100 - 18t \implies 18t \leq 100 \implies t \leq \frac{100}{18} \approx 5.56
     \]

   - Now, consider the upper bound \( 100 - 18t \leq 100 \):
     \[
     100 - 18t \leq 100 \implies 0 \leq 18t \implies t \geq 0
     \]

   - Combining these two results gives:
     \[
     0 \leq t \leq \frac{100}{18}
     \]

### Conclusion

The domain of the function \( B(t) = 100 - 18t \) is:
\[
[0, \frac{100}{18}] \quad \text{or approximately} \quad [0, 5.56]
\]

This means the function is valid for values of \( t \) ranging from 0 to approximately 5.56 years after Fall 2015. After about 5.56 years, the percentage of professors still using the old system would reach 0%, so the function is not defined beyond that point.

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Would you like more details on any part of this solution, or do you have any other questions?

### Related Questions

1. What is the value of \( B(t) \) exactly 3 years after Fall 2015?
2. How long will it take until only 50% of professors are still using the old system?
3. What is the rate of change of the percentage of professors using the old system per year?
4. If the function were modified to \( B(t) = 100 - kt \), how would the domain change?
5. How would you represent the situation graphically, and what does the slope represent?

**Tip:** Always check the context of a problem to determine reasonable domains and ranges for functions!

UNL has decided to gradually transition from one online gradebook system to another. The percentage of professors still using the old system years after Fall 2015 is given by B(t)=100-18t. Find domain.

Understand how to find the domain of the linear function B(t) = 100 - 18t, representing the percentage of professors using an old gradebook system years after Fall 2015. This problem explores key algebraic concepts such as domain, range, and linear inequalities. Suitable for students in Grades 9-12.

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