Sure, let's break this down step-by-step to estimate how many more students were full-time than part-time in 2012.
### Step 1: Identify the Value of [tex]\( x \)[/tex] for the Year 2012
Since [tex]\( x \)[/tex] represents the number of years since 2008, we calculate [tex]\( x \)[/tex] for 2012 as:
[tex]\[ x = 2012 - 2008 = 4 \][/tex]
### Step 2: Define the Functions for Full-Time and Part-Time Students
We have the following functions representing the number of full-time and part-time students in thousands:
- Full-time students function:
[tex]\[
F(x) = -2.634x^4 + 57.36x^3 - 444.5x^2 + 1258x + 10236
\][/tex]
- Part-time students function:
[tex]\[
P(x) = 6.34x^3 - 99.5x^2 + 432.7x + 6105
\][/tex]
### Step 3: Calculate the Number of Full-Time Students in 2012
Substitute [tex]\( x = 4 \)[/tex] into the full-time students function:
[tex]\[
F(4) = -2.634(4)^4 + 57.36(4)^3 - 444.5(4)^2 + 1258(4) + 10236
\][/tex]
Upon calculating the expression, the number of full-time students in thousands is:
[tex]\[
F(4) \approx 11152.736
\][/tex]
### Step 4: Calculate the Number of Part-Time Students in 2012
Substitute [tex]\( x = 4 \)[/tex] into the part-time students function:
[tex]\[
P(4) = 6.34(4)^3 - 99.5(4)^2 + 432.7(4) + 6105
\][/tex]
Upon calculating the expression, the number of part-time students in thousands is:
[tex]\[
P(4) \approx 6649.56
\][/tex]
### Step 5: Calculate the Difference in the Number of Students
To find out how many more students were full-time than part-time, we calculate the difference:
[tex]\[
\text{Difference} = F(4) - P(4)
\][/tex]
[tex]\[
\text{Difference} \approx 11152.736 - 6649.56 = 4503.176
\][/tex]
### Conclusion
Therefore, in 2012, there were approximately 4503.176 thousand more full-time students than part-time students.
