SL 3.1—3d space, volume, angles, distance, midpoints

The diameter is $8.4$ m, so the radius is $r=\frac{8.4}{2}=4.2$ m. The volume is $V=415$ m³.

The volume of a cylinder is $V=\pi r^{2}h$. Substituting: $415=\pi (4.2{)}^{2}h=17.64\pi h$

$h=\frac{415}{17.64\pi }=\frac{415}{55.418\dots }\approx 7.489\dots$

Rounding $7.489\dots$ to 3 significant figures gives $h\approx 7.48$ m.

We need $h=\frac{V}{\pi r^2}=\frac{415}{\pi (4.2{)}^2}$. The radius is $4.2$ m (half the diameter of $8.4$ m).

$14.9$ m would result from using diameter $8.4$ instead of radius $4.2$ in the formula (i.e. $r=8.4$), which is a common error. This gives approximately twice the correct answer.

$3.74$ m is approximately half of the correct answer. This could arise from incorrectly doubling $r^2$ in the denominator.

$18.7$ m does not match $\frac{415}{\pi (4.2{)}^2}$ under any standard substitution and is far too large.

The calculation $h=\frac{415}{\pi (4.2{)}^2}\approx 7.48$ m confirms $7.48$ m is correct.

$B=(0,2,4)$ and $C=(2k,0,k+1)$.

$\Delta x=2k-0=2k,\Delta y=0-2=-2,\Delta z=(k+1)-4=k-3$

$|BC{|}^2=(2k{)}^2+(-2{)}^2+(k-3{)}^2=4k^2+4+(k-3{)}^2$

$|BC|=\sqrt{4k^2+4+(k-3{)}^2}$

We need $|BC|=\sqrt{(2k-0{)}^2+(0-2{)}^2+(k+1-4{)}^2}$. Each coordinate difference must be squared correctly.

$\sqrt{4k^2-4k+9+(k-3{)}^2}$ expands the $y$-difference incorrectly, suggesting $\Delta y$ was treated as $2k-2$ instead of $-2$.

$\sqrt{4k^2+4+k^2}$ uses $k+1$ rather than $k-3$ for the $z$-difference — this forgets to subtract $B$'s $z$-coordinate of $4$.

$\sqrt{(2k{)}^2+(0-2{)}^2+(k+1-4{)}^2}$ is actually equivalent to the correct answer when expanded, but it is written in an unsimplified form with the middle term not evaluated — it is not the cleanest final expression.

$\sqrt{4k^2+4+(k-3{)}^2}$ correctly computes $(2k{)}^2+(-2{)}^2+(k-3{)}^2$ and is the clearest correct form.