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. 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:

a = 4 m v = 3 m s^{2} = (a / 2)^{2} + v^{2} s = (a / 2)^{2} + v^{2} = (4 / 2)^{2} + 3^{2} = 13 m ≐ 3.6056 m (v - h) : r = v : a / 2 v = h + r \cdot v : a / 2 h = v - r \cdot 2 v / a V = π r^{2} h V = π r^{2} (v - r \cdot 2 v / a) V = π r^{2} (v - r \cdot 2 v / a) V = π r^{2} (3 - r \cdot 2 \cdot 3 / 4) V^{'} = 3 / 2 π (a - v r) r V^{'} = 0 3 / 2 π \cdot (a - v r) r = 0 3 / 2 * p i * (4 - 3 r) r = 0 3 / 2 \cdot 3.1415926 \cdot (4 - 3 r) r = 0 - 14.137167 r^{2} + 18.85 r = 0 14.137167 r^{2} - 18.85 r = 0 a = 14.137167; b = - 18.85; c = 0 D = b^{2} - 4 a c = 18.8 5^{2} - 4 \cdot 14.137167 \cdot 0 = 355.3057463175 D > 0 r_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 8 . 8 5 \pm 3 5 5 . 3 1}{2 8 . 2 7 4 3 3 4} r_{1, 2} = 0.66666665 \pm 0.66666665251956 r_{1} = 1.3333333050391 r_{2} = 0 Factored form of the equation: 14.137167 (r - 1.3333333050391) r = 0 r = r_{1} = 1.3333 = 1.3333 m

Our quadratic equation calculator calculates it.

h = v - r \cdot 2 \cdot v / a = 3 - 1.3333 \cdot 2 \cdot 3 / 4 = 1 m

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