Answer
a) $h(t) = -0.0096(t-32)^2+32.5$
b) $32,000\ ft$.
c) Domain: $[0, 90]$
Range: $[0.21,32.5]$
Work Step by Step
a) Make a scatter plot of the data and choose a vertex point that may either be the lowest or highest point. The vertex of the data point can be found to be at $(h,k)=(32,32.5)$.
Set this into the standard vertex form of a parabola. This gives
$$\begin{aligned}
f(x) &= a(x-32)^2+32.5
\end{aligned}$$ Choose any point from the scatter plot to find the value of the constant $a$. Let's take the point $(x,y)=(60,25)$ and insert this into the above equation to find $a$. $$\begin{aligned}
25 &= a(60-32)^2+32.5\\
25& = 28^2a+32.5\\
25-32.5& =784a \\
784a= -7.5\\
a\approx -0.0096
\end{aligned}$$ Hence, the parabola that best fits the data is given by $$\begin{aligned}
h(t) &= -0.0096(t-32)^2+32.5,
\end{aligned}$$ where $t$ is the travel time of the turbojet, and $h(t)$ is its height in thousands of feet.
b) we determine $h(25)$: $$\begin{aligned}
h(25) &= -0.0096(25-32)^2+32.5= 32.0\\
\end{aligned}$$ The height of the plane after $25$ seconds is about $32,000\ ft$.
c) Assume that we are measuring the height of the plane from $0$ seconds to about $90$ seconds. Then: $$\begin{aligned}
h(0) &= -0.0096(0-32)^2+32.5= 22.7\\
h(90)&= -0.0096(90-32)^2+32.5= 0.21.
\end{aligned}$$ The domain and range would be:
Domain: $[0, 90]$
Range: $[0.21, 32.5]$
