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:

$a = 4 \ m \ \\ v = 3 \ m \ \\ \ \\ s^2 = (a/2)^2 + v^2 = (4/2)^2 + 3^2 = 13 \ \\ s = \sqrt{ (a/2)^2 + v^2 } = \sqrt{ (4/2)^2 + 3^2 } = \sqrt{ 13 } \ m \doteq 3.6056 \ m \ \\ \ \\ (v-h):r = v:a/2 \ \\ v = h + r \cdot \ v:a/2 \ \\ h = v - r \cdot \ 2v/a \ \\ \ \\ V = \pi r^2 \ h \ \\ V = \pi r^2 (v-r \cdot \ 2v/a) \ \\ V = \pi r^2 (v-r \cdot \ 2v/a) \ \\ V = \pi r^2 (3-r \cdot \ 2 \cdot \ 3/4) \ \\ V' = 3/2 \pi (a-vr)r \ \\ V' = 0 \ \\ \ \\ 3/2 \pi \cdot \ (a-vr)r = 0 \ \\ \ \\ \ \\ 3/2 \cdot \ 3.1415926 \cdot \ (4-3r)r = 0 \ \\ -14.1371667r^2 +18.85r = 0 \ \\ 14.1371667r^2 -18.85r = 0 \ \\ \ \\ a = 14.1371667; b = -18.85; c = 0 \ \\ D = b^2 - 4ac = 18.85^2 - 4\cdot 14.1371667 \cdot 0 = 355.305746317 \ \\ D>0 \ \\ \ \\ r_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 18.85 \pm \sqrt{ 355.31 } }{ 28.2743334 } \ \\ r_{1,2} = 0.66666667 \pm 0.666666666667 \ \\ r_{1} = 1.33333333333 \ \\ r_{2} = 0 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ r=14.1371667 (r -1.33333333333) r = 0r = r_{ 1 } = 1.3333 = \dfrac{ 4 }{ 3 } \doteq 1.3333 = 1.333 \ \text { m }$

Checkout calculation with our calculator of quadratic equations.

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

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