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.

Result

r =  1.333 m
h =  1.001 m

Solution:

a=4 m v=3 m  s2=(a/2)2+v2 s=(a/2)2+v2=(4/2)2+32≐13 m≐3.6056 m  (v−h):r=v:a/2 v=h+r⋅ v:a/2 h=v−r⋅ 2v/a  V=πr2 h V=πr2(v−r⋅ 2v/a) V=πr2(v−r⋅ 2v/a) V=πr2(3−r⋅ 2⋅ 3/4) V′=3/2π(a−vr)r V′=0  3/2π⋅ (a−vr)r=0  3/2∗pi∗(4−3r)r=0  3/2⋅ 3.1415926⋅ (4−3r)r=0 −14.1371667r2+18.85r=0 14.1371667r2−18.85r=0  a=14.1371667;b=−18.85;c=0 D=b2−4ac=18.852−4⋅14.1371667⋅0=355.305746317 D>0  r1,2=−b±D2a=18.85±355.3128.2743334 r1,2=0.66666667±0.666666666667 r1=1.33333333333 r2=0   Factored form of the equation:  14.1371667(r−1.33333333333)r=0 r=r1=1.3333≐43≐1.3333≐1.333 ma=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 } \doteq \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'=3/2 \pi (a-vr)r \ \\ V'=0 \ \\ \ \\ 3/2 \pi \cdot \ (a-vr)r=0 \ \\ \ \\ 3/2 * pi *(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=14.1371667; 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}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 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 \doteq \dfrac{ 4 }{ 3 } \doteq 1.3333 \doteq 1.333 \ \text{m}

Checkout calculation with our calculator of quadratic equations.

h=v−r⋅ 2⋅ v/a=3−1.3333⋅ 2⋅ 3/4=1.0005=1.001 mh=v - r \cdot \ 2 \cdot \ v/a=3 - 1.3333 \cdot \ 2 \cdot \ 3/4=1.0005=1.001 \ \text{m}

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