Secret treasure

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

Correct result:

r =  1.3333 m
h =  1 m

Solution:

$$\begin{gathered} a=4\text{ m } \\ v=3\text{ m } \\ {s}^2=(a\mathrm{/}2{)}^2+v^2 \\ s=\sqrt{(a\mathrm{/}2{)}^2+v^2}=\sqrt{(4\mathrm{/}2{)}^2+3^2}=\sqrt{13}\text{ m}\doteq 3.6056\text{ m } \\ (v-h):r=v:a\mathrm{/}2 \\ v=h+r\cdot v:a\mathrm{/}2 \\ h=v-r\cdot 2v\mathrm{/}a \\ V=\pi r^{2}h \\ V=\pi r^2(v-r\cdot 2v\mathrm{/}a) \\ V=\pi r^2(v-r\cdot 2v\mathrm{/}a) \\ V=\pi r^2(3-r\cdot 2\cdot 3\mathrm{/}4) \\ {V}^{\mathrm{\prime }}=3\mathrm{/}2\pi (a-vr)r \\ {V}^{\mathrm{\prime }}=0 \\ 3\mathrm{/}2\pi \cdot (a-vr)r=0 \\ 3\mathrm{/}2\ast pi\ast (4-3r)r=0 \\ 3\mathrm{/}2\cdot 3.1415926\cdot (4-3r)r=0 \\ -14.1371667r^2+18.85r=0 \\ 14.1371667r^2-18.85r=0 \\ a=14.1371667;b=-18.85;c=-0 \\ D=b^2-4ac=18.85^2-4\cdot 14.1371667\cdot -0=355.305746317 \\ D>0 \\ {r}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{18.85\pm \sqrt{355.31}}{28.2743334}} \\ {r}_{1,2}=0.66666667\pm 0.666666666667 \\ {r}_1=1.33333333333 \\ {r}_2=0 \\ \text{ Factored form of the equation: } \\ 14.1371667(r-1.33333333333)r=0 \\ r=r_1=1.3333={\displaystyle \frac{4}{3}}=1.3333\text{ m} \end{gathered}$$

Our quadratic equation calculator calculates it.

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

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