Question

question 8 (mandatory) (1 point) saved
the population of a village can be modelled by the function $p(x) = -22.5x^2 + 428x + 1100$, where $x$ is the number of years since 1990. according to the model, when will the population be the highest?
a) 1999
b) 2005
c) 1991
d) 2002

Explanation:

Step1: Recall vertex formula for parabola

For a quadratic function \( P(x) = ax^2 + bx + c \) (where \( a
eq 0 \)), the x - coordinate of the vertex (which gives the maximum or minimum value) is given by \( x = -\frac{b}{2a} \). In the function \( P(x)=-22.5x^{2}+428x + 1100 \), we have \( a=-22.5 \) and \( b = 428 \).

Step2: Calculate the x - coordinate of the vertex

Substitute \( a=-22.5 \) and \( b = 428 \) into the formula \( x=-\frac{b}{2a} \).
\( x=-\frac{428}{2\times(-22.5)}=-\frac{428}{-45}\approx9.51 \)

Step3: Determine the year

Since \( x \) is the number of years since 1990, we add \( x\approx9.51 \) to 1990. \( 1990 + 9.51\approx1999.51 \), which is closest to 1999.

Answer:

a) 1999