To determine how many more students were full-time than part-time in 2012, we can use the given models for the number of full-time and part-time students.
The functions provided are:
[tex]\[ F(x) = -2.634x^4 + 57.36x^3 - 444.5x^2 + 1258x + 10,236 \][/tex]
[tex]\[ P(x) = 6.34x^3 - 99.5x^2 + 432.7x + 105 \][/tex]
Here, [tex]\( F(x) \)[/tex] represents the number of full-time students in thousands and [tex]\( P(x) \)[/tex] represents the number of part-time students in thousands. The variable [tex]\( x \)[/tex] denotes the number of years since 2008.
1. Determine [tex]\( x \)[/tex] for the year 2012:
Since [tex]\( x \)[/tex] is the number of years since 2008,
[tex]\[ x = 2012 - 2008 = 4 \][/tex]
2. Calculate the number of full-time students in 2012:
[tex]\[ F(4) = -2.634(4)^4 + 57.36(4)^3 - 444.5(4)^2 + 1258(4) + 10,236 \][/tex]
By evaluating this expression, the number of full-time students is approximately:
[tex]\[ F(4) \approx 11,152.736 \; \text{thousands} \][/tex]
3. Calculate the number of part-time students in 2012:
[tex]\[ P(4) = 6.34(4)^3 - 99.5(4)^2 + 432.7(4) + 105 \][/tex]
By evaluating this expression, the number of part-time students is approximately:
[tex]\[ P(4) \approx 649.56 \; \text{thousands} \][/tex]
4. Determine how many more students were full-time than part-time:
To find the difference between the number of full-time and part-time students,
[tex]\[ \text{Difference} = F(4) - P(4) \][/tex]
Substituting the values we found:
[tex]\[ \text{Difference} = 11,152.736 - 649.56 \][/tex]
[tex]\[ \text{Difference} \approx 10,503.176 \; \text{thousands} \][/tex]
Thus, in 2012, there were approximately 10,503.176 thousand more full-time students than part-time students.
