Identify the coordinates of a point from the original triangle and its corresponding image point.

Let's use point X and its image X'.
From the graph, the coordinates are:
$X = (3, 0)$
$X' = (0, 3)$

A translation vector $M = \begin{pmatrix} a \\ b \end{pmatrix}$ maps a point $(x, y)$ to $(x+a, y+b)$.
So, $X' = X + M$.
To find M, we subtract the original point's coordinates from the image point's coordinates:

The translation vector is $\boxed{\begin{pmatrix} -3 \\ 3 \end{pmatrix}}$.

17. A quantity P varies directly as q and inversely as the cube root of r. When p = 10 and q = 5 then r = 8. Find P when q = 54 and r = 27.

"P varies directly as q" means $P \propto q$.
"P varies inversely as the cube root of r" means $P \propto \frac{1}{\sqrt[3]{r}}$.
Combining these, we get the relationship:

where $k$ is the constant of proportionality.

Given $P = 10$, $q = 5$, and $r = 8$.
Substitute these values into the equation:

Calculate the cube root of 8: $\sqrt[3]{8} = 2$.

Multiply both sides by 2:

Divide by 5:

Substitute the value of $k$ back into the equation:

Substitute $q = 54$ and $r = 27$ into the complete variation equation:

Calculate the cube root of 27: $\sqrt[3]{27} = 3$.

The value of P is $\boxed{72}$.