I've heard about the following result: for each two natural numbers $d,n\in\mathbb{N}$ one can find a number $k\in\mathbb{N}$ with the following property: for each CW-complex $X$ with $\dim X\leq d$ and a rank $n$ vector bundle $E\to X$ there is a continuos map into the Grassmanian $f:X\to G_n(\mathbb{R}^{n+k})$ such that $E\simeq f^*\gamma_n\left(\mathbb{R}^{n+k}\right)$ where $\gamma_n\left(\mathbb{R}^{n+k}\right)$ is the tautological rank $n$ vector bundle over $G_n(\mathbb{R}^{n+k})$.
Question 1 Could anybody give me reference to this theorem such that in the proof it is apparent how $k$ depends of $n,d$?
Question 2 if $n,k$ are given numbers, how many trivializations are needed for covering $\gamma_n\left(\mathbb{R}^{n+k}\right)$?
Of course, please let me know if the above formulation needs some corrections.