Answer
$H=81l^{2}w^{2}$
Work Step by Step
$H$ is jointly proportional to $l^2$ and $w^2$ so the equation that relates the variables is:
$H=k(l^2w^2wh)$ where k = constant of proportionality.
If l = 2 and $w = \frac{1}[{3}$, then H = 36. Substitute these values into the equation above to have:
$H=k(l^2w^2)
\\36 = k(2^2 \cdot (\frac{1}{3})^2)
\\36=k(4 \cdot \frac{1}{9})
\\36 = k \cdot \frac{4}{9}
\\\frac{9}{4} \cdot 36 = k
\\81=k$
The constant of proportionality is $81$. Thus, the equation that relates H, l, and w is:
$H=81(l^2w^2)$
