Determine image of hypercube under linear map

Question

Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb R^N$ with each entry of $w$ in the range $[0,1]$. In other words I'd like to find the image of the unit hypercube in $\Bbb R^N$ under the map represented by $A$. Does there exist an algorithm that runs in time polynomial in $N$?

Everything I've thought of so far is exponential, basically looping over the $2^N$ vertices of the hypercube. The problem is similar to that of determining the convex hull of a given set of points, except that the entries in $w$ needn't sum to $1$. Obviously taking the convex hull of all $2^N$ points would work, but it's exponential again.

My motivation is that I want to determine the range of colours that can be seen under a given light source, based on the spectral power distribution of the light source and the responsivity curves of the three cone cells in the eye.