To find the approximate value of $\sqrt{19}$ using a number line, we find the two perfect squares that 19 is between. Since $4^2=16<19$ and $5^2=25>19$, 19 is between the perfect squares 16 and 25. On a number line with 0 at the origin, we place a mark at 16 and a mark at 25. Here is a number line with marks at 16 and 25: [asy] size(200); int i; for(i=0;i<=4;++i) { draw((4*i,-2)--(4*i,2), linewidth(0.7)); label("$" + string(4*i) + "$", (4*i,-2), S); } label("$0$",(0,-2),S); draw((-2,0)--(25,0), Arrows); draw((16,-1.5)--(16,1.5), linewidth(0.6)+blue); draw((25,-1.5)--(25,1.5), linewidth(0.6)+blue); dot((16,0)); dot((25,0)); label(rotate(270)*"$16$",(16,0),S); label(rotate(270)*"$25$",(25,0),S); [/asy] The point representing 19 is approximately $\frac{3}{4}$ of the way between 16 and 25. We draw a dashed line to approximate this distance as follows: [asy] size(200); int i; for(i=0;i<=4;++i) { draw((-2,4*i)--(25,4*i), linewidth(0.7)); label("$" + string(4*i) + "$", (-2,4*i), W); } label("$0$",(-2,0),W); draw((-2,0)--(25,0), Arrows); draw((16,-2)--(16,10), linewidth(0.6)+blue); draw((25,-2)--(25,10), linewidth(0.6)+blue); draw((16,-0.65)--(25,4.6), linewidth(0.6)+red+ dashed); dot((16,0)); dot((25,0)); label(rotate(270)*"$16$",(16,0),S); label(rotate(270)*"$25$",(25,0),S); [/asy] Therefore, $\sqrt{19}$ is approximately $\boxed{4.6}$.
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