Answer
a) $0.123$ grams
b) $0.176 $ grams
Work Step by Step
Given \begin{equation}
V(m)=17.6 \sqrt[5]{m^3}= 17.6\left(m \right)^{\frac{3}{5}}.
\end{equation} First solve for $M$ in term of $V$.
\begin{equation}
\begin{aligned}
17.6\left(m \right)^{\frac{3}{5}}&=V\\
\left(m \right)^{\frac{3}{5}}&=\frac{V}{17.6 } \\
\left(\left(m \right)^{\frac{3}{5}}\right)^{\frac{5}{3}}& =\left( \frac{V}{17.6 }\right)^{\frac{5}{3}}\\
m& = \left( \frac{V}{17.6 }\right)^{\frac{5}{3}}\\
m& = \sqrt[3]{\left( \frac{V}{17.6 }\right)^5}.
\end{aligned}
\end{equation} a) Now, set $V= 5$ to find the thoracic mass of a butterfly that can fly at $5$ meters per second. \begin{equation}
\begin{aligned}
m& = \sqrt[3]{\left( \frac{5}{17.6 }\right)^5}\\
&=0.123.
\end{aligned}
\end{equation} The thoracic mass of a butterfly that can fly at $5$ meters per second is about $0.123$ grams.
b) Set $V= 6.2$ to find the thoracic mass of a butterfly that can fly at $6.2$ meters per second. \begin{equation}
\begin{aligned}
m& = \left( \frac{6.2}{17.6 }\right)^{\frac{5}{3}}\\
&=0.176.
\end{aligned}
\end{equation} The thoracic mass of a butterfly that can fly at $5$ meters per seconds is about $0.176 $ grams.
