Question

the table shows the population of a town from 1996 to 2004. assume that t is the number of years since 1996 and p is measured in thousands of people. which of the following is the model that best fits the data rounded to the nearest tenth? ( p(t)=-.2x^{2}+2.1x + 22.9 ) ( p(t)=-.2x^{2}+2x + 22.9 ) ( p(t)=-0.21x^{2}+2.08x + 22.96 ) ( p(t)=-.2x^{2}+2.1x + 23.0 )

Explanation:

Step1: Substitute \(t = 0\) into each model

• For \(P(t)=-0.2t^{2}+2.1t + 22.9\), when \(t = 0\), \(P(0)=-0.2\times0^{2}+2.1\times0 + 22.9=22.9\)
• For \(P(t)=-0.2t^{2}+2t + 22.9\), when \(t = 0\), \(P(0)=-0.2\times0^{2}+2\times0 + 22.9=22.9\)
• For \(P(t)=-0.21t^{2}+2.08t + 22.96\), when \(t = 0\), \(P(0)=-0.21\times0^{2}+2.08\times0 + 22.96=22.96\)
• For \(P(t)=-0.2t^{2}+2.1t + 23.0\), when \(t = 0\), \(P(0)=-0.2\times0^{2}+2.1\times0 + 23.0=23.0\)

Since the population in 1996 (\(t = 0\)) is \(22.8\) (measured in thousands), models with \(P(0)=22.9\) are closer.

Step2: Check another value, say \(t = 1\)

• For \(P(t)=-0.2t^{2}+2.1t + 22.9\), \(P(1)=-0.2\times1^{2}+2.1\times1 + 22.9=- 0.2+2.1 + 22.9=24.8\)
• For \(P(t)=-0.2t^{2}+2t + 22.9\), \(P(1)=-0.2\times1^{2}+2\times1 + 22.9=-0.2 + 2+22.9=24.7\)

The actual value at \(t = 1\) is \(25.0\). The first model \(P(t)=-0.2t^{2}+2.1t + 22.9\) gives \(24.8\) which is closer compared to \(24.7\) (when \(t = 1\))

Answer:

\(P(t)=-0.2t^{2}+2.1t + 22.9\)