Answer
a) $\$1962.2$ million
b) $1999$, $2003$
c) $2001$
Work Step by Step
Given \begin{equation}
I(t)=-145.86 t^2+3169.342 t-15145.2.
\end{equation} a) Set $t= 10$ to estimate the net income for Dell in the year $2000$.
\begin{equation}
\begin{aligned}
I(10) & =-145.86 \cdot 10^2+3169.34 \cdot 10-15145.2 \\
& =1962.2
\end{aligned}
\end{equation} The net income for Dell in $2000$ was about $\$ 1962.2$ million.
b) Set $I(t)= 1500$ to find the values of $t$, which gives the year(s) that the sale will reach $\$1500$ million .
\begin{equation}
\begin{aligned}
-145.86 t^2+3169.342 t-15145.2 & =1500 \\
-145.86 t^2+3169.342-15145.2-1500 & =0 \\
-145.86 t^2+3169.342 t-16645.2 & =0.
\end{aligned}
\end{equation} Solve the equation
$$ \begin{aligned}
t & =\frac{-3169.34 \pm \sqrt{3169.34^2-4(-145.86)(-16645.2)}}{2(-145.86)} \\
& =-\frac{-3169.34 \pm \sqrt{333240.5476}}{291.72} \\
& =-(-10.8643 \pm 1.9788).
\end{aligned}
$$ The solutions are: $$
\begin{aligned}
t_1 & =-(-10.8643+1.9788) \\
& \approx 8.89 \\
t_2 & =-(-10.8643+1.9788) \\
& \approx 12.84.
\end{aligned}
$$ Dell's net income reached $1500$ million in about $1999$ and again in about $2003$.
c) The vertex of the income function will give us the maximum income. Use $a= -145.86$ and $b= -3169.342$ into the following formula. $$
\begin{aligned}
t & =\frac{-b}{2 a} \\
& =\frac{-3169.342}{2(-145.86)} \\
& =10.864
\end{aligned}
$$ $$
\begin{aligned}
I_{max} & =-145.86(10.864)^2+3169.342(10.364)-15145.2 \\
& =2071.187.
\end{aligned}
$$ The vertex is $(10.864,2071.187)$. This means that the company had the highest income of $\$2071.187$ million in $2001$.
