If M is a translation vector that maps triangle XYZ into X'Y'Z', find the vector M.

From the graph, the coordinates are:
Original point $X = (3, 0)$
Image point $X' = (1, 3)$

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

I've also checked with Y and Y' ($Y=(2,-1)$, $Y'=(0,2)$) and Z and Z' ($Z=(5,-3)$, $Z'=(3,0)$), and they all consistently give the same translation vector.
The translation vector is $\boxed{\begin{pmatrix} -2 \\ 3 \end{pmatrix}}$.

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.

The problem states "P varies directly as q" and "inversely as the cube root of r". This can be written as:

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 $\boxed{72}$.