Answer
a) $59.20^{\circ}F$ .
b) July
c) June and September
Work Step by Step
Given \begin{equation}
H(m)=-1.5 m^2+21.6 m-3.2.
\end{equation} a) Set $m=4$ to find the average temperature for the month of June.
\begin{equation}
\begin{aligned}
H(6) & =-1.5 \cdot 4^2+21.6\cdot 4-3.2 \\
& =59.20.
\end{aligned}
\end{equation} This is the average temperature in Paris for the month of April which is about $59.20^{\circ}F$.
b) Use the formula, with $a= -1.5, b= 21.6$ to find the vertex.
\begin{equation}
\begin{aligned}
m&=\frac{-b}{2a} \\
& =\frac{-(21.6)}{2(-1.5)} \\
& = 7.2 \\
H(7.2)&=-1.5 \cdot 7.2^2+21.6\cdot 7.2-3.2\\
& =74.56
\end{aligned}
\end{equation} The vertex of the function is $(7.2,74.56)$.This means that for the month of July, the average temperate in Paris is about $74.56^{\circ}F$ which is the maximum temperature of the city.
c) Set $H=70$ to find the value(s) of $m$.
\begin{equation}
\begin{aligned}
-1.5 m^2+21.6 m-3.2& =70 \\
-1.5 m^2+21.6 m-3.2-70& =0 \\
-1.5 m^2+21.6 m-73.2& =0 \\
\left( -1.5 m^2+21.6 m-73.2\right)\cdot (-10) & =0 \\
15m^2-216m+732& =0
\end{aligned}
\end{equation} Solve the equation: \begin{equation}
\begin{aligned}
&m= \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
& m=\frac{-(-216) \pm \sqrt{(-216)^2-4(15)(732)}}{2(15)} \\
& =\frac{216 \pm \sqrt{2736}}{18}\\
&= \frac{216\pm 52.3068}{18}\\
&= 7.2\pm 1.744\\
\end{aligned}
\end{equation} The solutions are: \begin{equation}
\begin{aligned}
& m=7.2- 1.744\approx 5.46 \\
& m=7.2+ 1.744\approx 8.94.
\end{aligned}
\end{equation} The average temperature in Paris is $70\ ^{\circ}F$ during the months of June and September.
