Question

A group of aliens invade earth to harvest lava from a volcano. Assuming that the volcano is in the shape of a cone of radius 2000 meters and a height of 1000 meters, find the amount of work it will take to pump all of the lava through the top of the volcano (assuming it’s entirely full of lava). Ignore ρ and g in your answer.

Answer 1

Work done is the product of the force and the distance moved in the force's direction

The amount of work it would take to pump all the lava through the top of the volcano is π × 10¹²

The reason the above value is correct is given as follows:

The radius of the cone, r = 2,000 m

The height of the cone, h = 1,000 m

Considering a slice of volume of the cone, we have;

Volume of slice, $V_i$ = π·$r_i^2$·Δx

Where;

$r_i$ = Radius of slice

Δx = Height of slice

Considering similar triangles ΔABC and ΔADE, we have;

$\dfrac{r_i}{x_i} = \dfrac{2,000}{1,000} = 2$

$r_i = 2 \cdot x_i$

$V_i$ = π·$r_i^2$·Δx

∴ $V_i$ = π·(2·$x_i$)²·Δx = 4·π·$x_i$²·Δx

Work done = F × d

Where;

d = Distance = $x_i$

Work done = 4·π·$x_i$²·Δx × $x_i$

$\displaystyle \int\limits^{1000}_0 {4 \cdot \pi \cdot x_i^3 \cdot } \, dx = \left[4 \times \pi \times \frac{x_i^4}{4} \right]_0^{1000}=\pi \times 1000^4 = \pi \times 10^{3 \times 4} = \pi\times 10^{12}$

The work done, W = π × 10¹²

Learn more about work done by pumping fluid out of a vessel here:

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