Answer
a) $ R(p) = -(p-20)^2+500$
b) Vertex: $(20,500)$
c) $\$ 496$
d) Domain: $[6, 35]$
Range: $[275, 500]$
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 $(h,k)=(20,500)$.
Set this into the standard vertex form of a parabola. This gives $$\begin{aligned}
f(x) &= a(x-20)^2+500\\
\end{aligned}$$ Choose any point from the scatter plot to find the value of the constant $a$. Let's take the point $(x,y)=(10,400)$ and insert this into the above equation to find $a$. $$\begin{aligned}
400 &= a(10-20)^2+500\\
400& = 10^2a+500\\
400-500& =100a \\
100a= -100\\
a= -1.
\end{aligned}$$ Hence, the parabola that best fit the data is the revenue function. It is given by $$\begin{aligned}
R(p) &= -(p-20)^2+500,
\end{aligned}$$ where $p$ is the selling price of each cycling glove and $R(p)$ is the total revenue generated from selling cycling gloves.
b) The vertex gives the maximum revenue of $\$500$ that is achieved from sell each pair of gloves for $\$ 20$.
c) We determine $R(22)$: $$\begin{aligned}
R(22) &= -(22-20)^2+500\\
& = \$496.
\end{aligned}$$ The revenue for sell each pair of gloves for $\$22$ is $\$ 496$.
d) Assume that the minimum and maximum price that each pair gloves can be sold for while still making profit are $\$ 6$ and $\$ 35$ respectively. Then: $$\begin{aligned}
R(6) &= -(6-20)^2+500=304 \\
R(35) &= -(35-20)^2+500= 275.
\end{aligned}$$ The domain and range are:
Range: $[6, 35]$
Domain: $[275, 500]$
