Determine the average speed of a car between an hour and a half and a half hour after it starts a trip when its distance-time function is $d(t)=t^3-15t^2+75t$.

To find the average speed we need to do the following:
\[
\mbox{Average Speed}=\frac{d(1.5)-d(0.5)}{1.5-0.5}.
\]
Doing the algebra we find that
\[
d(1.5)=(1.5)^3-15(1.5)^2+75(1.5)=82.125
\]
and
\[
d(0.5)=(0.5)^3-15(0.5)^2+75(0.5)=33.875.
\]

Now we find that
\begin{eqnarray}
\mbox{Average Speed}&=&\frac{d(1.5)-d(0.5)}{1.5-0.5}\\
&=&\frac{82.125-33.875}{1}\\
&=&48.25
\end{eqnarray}
So the average speed is $48.25$ miles per hour.