Answer
$\color{blue}{T=d^2\sqrt[3]{x}}$
Work Step by Step
RECALL:
(1) If $y$ varies directly as $x$, then $y=kx$ where $k$ is the constant of proportionality.
(2) If $y$ varies inversely as $x$, then $y=\dfrac{k}{x}$ where $k$ is the constant of proportionality.
(3) If $y$ varies jointly as $x$ and $z$, then $y = kxz$ where $k$ is the constant of proportionality.
Notice that when the variation is direct or joint, the variable/ are on the numerator while if the variation is inverse, the variable is in the denominator.
$T$ varies jointly with the cube root of $x$ and the square of $d$. Using the formula in (3) above, the equation of the joint variation is:
$T=k\cdot \sqrt[3]{x} \cdot d^2
\\T=kd^2\sqrt[3]{x}$
Since $T=18$ when $x=8$ and $d=3$, substituting these into the tentative equation above gives:
$\require{cancel}
T=kd^2\sqrt[3]{x}
\\18=k(3^2)\sqrt[3]{8}
\\18=k(9)(2)
\\18=k(18)
\\\frac{18}{18}=k
\\1=k$
Thus, the equation of the joint variation is:
$T=1\cdot d^2\sqrt[3]{x}
\\\color{blue}{T=d^2\sqrt[3]{x}}$
