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GATE 2019 CE (CE2) – Question 37

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 2 marks · Multiple choice

The hemispherical tank shown has radius 13 m. Find water volume (m³) when central water depth is 6 m.

Diagram for GATE 2019 CE (CE2) question 37
  1. $78\pi$
  2. $156\pi$
  3. $396\pi$
  4. $468\pi$

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Correct answer: (C) $396\pi$

Explanation

At height z above the hemisphere bottom, slice radius squared is $2Rz-z^2$. Integrate: $$V=\pi\int_0^h(2Rz-z^2)\,dz=\pi h^2(R-h/3).$$ For R = 13 m and h = 6 m, $V=36\pi(13-2)=$ **396π m³, C**.