Question

the number of admissions to all types of hospitals in a country between 1987 and 1998 can be described by the function ( a(t)=30.497t^{2}-1030.71t + 40,735.173 ) thousand people where ( t ) is the number of years after 1982. use the model and graphical methods to find in what year after 1982 the number of admissions was equal to 32,027,000. after 1982, the year in which the number of admissions was equal to 32,027,000 is ( square )

Explanation:

Step1: Set up the equation

We are given \(A(t)=30.497t^{2}-1030.71t + 40735.173\) (in thousands) and \(A(t)=32027\) (since \(32027000\div1000 = 32027\)). So the equation is \(30.497t^{2}-1030.71t+40735.173 = 32027\).
Subtract \(32027\) from both sides: \(30.497t^{2}-1030.71t + 40735.173-32027=0\), which simplifies to \(30.497t^{2}-1030.71t + 8708.173 = 0\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here \(a = 30.497\), \(b=- 1030.71\), and \(c = 8708.173\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-1030.71)^{2}-4\times30.497\times8708.173\).
\(\Delta = 1062363.1041-4\times30.497\times8708.173\)
\(\Delta=1062363.1041 - 1062363.1041=0\) (approximate calculation).
Then \(t=\frac{-(-1030.71)\pm\sqrt{0}}{2\times30.497}=\frac{1030.71}{60.994}\approx17\).

Answer:

\(1982 + 17=1999\)