Question

question 16 (3 points)
the table shows data from 1975 and projections to 2050 of the number of millions of people age 15 to 59 in a large country.

year	labor pool (millions)	year	labor pool (millions)
1980	580	2020	919
1985	654	2025	902
1990	730	2030	876
1995	759	2035	828
2000	803	2040	824
2005	870	2045	804
2010	906	2050	678

the quadratic function that best models this population as a function of the number of years after 1970 is given by $y = -0.219x^2 + 21.632x + 376.687$, where $x$ represents the number of years after 1970 and $y$ represents the number of millions of people in this labor pool.
using the model, determine the maximum size of this population before it begins to shrink.
the maximum size of the population is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ million.
(round up to the nearest integer as needed)
\bigcirc 912
\bigcirc 910
\bigcirc 908
\bigcirc 911

Explanation:

Step1: Recall vertex formula for quadratic

For \( y = ax^2 + bx + c \), vertex \( x \)-coordinate is \( x = -\frac{b}{2a} \). Here, \( a = -0.219 \), \( b = 21.632 \).
\( x = -\frac{21.632}{2(-0.219)} = \frac{21.632}{0.438} \approx 49.388 \)

Step2: Substitute x into quadratic

Plug \( x \approx 49.388 \) into \( y = -0.219x^2 + 21.632x + 376.687 \).
\( y = -0.219(49.388)^2 + 21.632(49.388) + 376.687 \)
Calculate \( (49.388)^2 \approx 2439.17 \)
\( -0.219(2439.17) \approx -534.18 \)
\( 21.632(49.388) \approx 1068.34 \)
Sum: \( -534.18 + 1068.34 + 376.687 \approx 910.847 \)

Step3: Round up to nearest integer

\( 910.847 \) rounded up is \( 911 \).

Answer:

911