Answer
a) $ R(p) = -85(p-8)^2+1440$
b) $(8,1440)$
c) $\$ 1248.8$
d) Domain: $[5, 10]$
Range: $[675, 1440]$
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 $(h,k)=(8,1440)$.
Set this into the standard vertex form of a parabola. This gives
$$\begin{aligned}
f(x) &= a(x-8)^2+1440.
\end{aligned}$$ Choose any point from the scatter plot to find the value of the constant $a$. Let's take the point $(x,y)=(10,1100)$ and insert this into the above equation to find $a$. $$\begin{aligned}
1100 &= a(10-8)^2+1440\\
1100& = 2^2a+500\\
1100-1440& =4a \\
4a= -340\\
a= -85.
\end{aligned}$$ Hence, the parabola that best fits the data is the revenue function. It is given by $$\begin{aligned}
R(p) &= -85(p-8)^2+1440,
\end{aligned}$$ where $p$ is the selling price of each water bottle and $R(p)$ is the total revenue generated from the selling of water bottles .
b) The vertex gives the maximum revenue of $\$1440$ that is achieved from sell each bottle of water for $\$ 8$.
c) We determine $R(9.5)$: $$\begin{aligned}
R(9.5) &= -(9.5-8)^2+1440\\
.& = \$1248.8
\end{aligned}$$ The revenue for selling each bottle of water for $\$9.5$ is $\$ 1248.8$.
d) Assume that the minimum and maximum price that each pair gloves can be sold for while still making profit are $\$ 5$ and $\$ 10$ respectively. Then: $$\begin{aligned}
R(5) &= -85(5-8)^2+1440= 675 \\
R(10) &= -85(10-8)^2+14400= 1100.
\end{aligned}$$ Domain: $[5, 10]$
Range: $[675, 1440]$
