Secret treasure

Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base of 4 m and a height of 3 m. Find the container's radius r (and height h) so that they can hide the largest possible treasure.

Correct answer:

r =  1.3333 m
h =  1 m

Step-by-step explanation:

$$\begin{gathered} a=4\text{m} \\ v=3\text{m} \\ {s}^2=(a/2{)}^2+v^2 \\ s=\sqrt{(a/2{)}^2+v^2}=\sqrt{(4/2{)}^2+3^2}=\sqrt{13}\text{m}\doteq 3.6056\text{m} \\ (v-h):r=v:a/2 \\ v=h+r\cdot v:a/2 \\ h=v-r\cdot 2v/a \\ V=\pi r^{2}h \\ V=\pi r^2(v-r\cdot 2v/a) \\ V=\pi r^2(v-r\cdot 2v/a) \\ V=\pi r^2(3-r\cdot 2\cdot 3/4) \\ {V}^{\prime }=3/2\cdot \pi (a-vr)r \\ {V}^{\prime }=0 \\ 3/2\pi \cdot (a-vr)r=0 \\ 3/2\cdot pi\cdot (4-3r)r=0 \\ 3/2\cdot 3.1415926\cdot (4-3r)r=0 \\ -14.1371667r^2+18.85r=0 \\ 14.1371667r^2-18.85r=0 \\ a=14.137167;b=-18.85;c=-0 \\ D=b^2-4ac=18.85^2-4\cdot 14.137167\cdot -0=355.3057463175 \\ D>0 \\ {r}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{18.85\pm \sqrt{355.31}}{28.274333} \\ {r}_{1,2}=0.666667\pm 0.666667 \\ {r}_1=1.333333333 \\ {r}_2=0 \\ r=r_1=1.3333=1.3333\text{m} \end{gathered}$$

Our quadratic equation calculator calculates it.

$h=v-r\cdot 2\cdot v/a=3-1.3333\cdot 2\cdot 3/4=1\text{m}$

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