Question

point) A tank in the shape of an inverted right circular cone has height 5 meters and radius 4 meters. It is filled with 4 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is δ=1080 kg/m3. Your answer must include the correct units.

Answer 1

Answer:

W = 907963.50 J = 907.96 J

Explanation:

Note: Refer to the figure attached

Now, from the figure we have similar triangles ΔAOB and ΔCOD

we have

$\frac{5}{4}=\frac{x}{r}$

or

$r=\frac{4x}{5}$

Now, the work done to empty the tank can be given as:

$W = \int\limits^4_0 {(5-x)\rho\times g A} \, dx$

or

$W = \int\limits^4_0 {(5-x)1080\times 9.8 (\pi r^2)} \, dx$

or

$W = \int\limits^4_0 {(5-x)\times10584\times (\pi (\frac{4x}{5})^2)} \, dx$

or

$W = 6773.76\pi\int\limits^4_0 {(5-x)x^2)} \, dx$

or

$W = 6773.76\pi[\frac{5}{3}x^3-\frac{1}{4}x^4]^4_0$

or

$W = 6773.76\pi[\frac{128}{3}]$

or

W = 907963.50 J = 907.96 J

Answer 2

The work required to empty the tank by pumping the hot chocolate over the top of the tank is 907.96 J.

What is work done?

Work done is the force applied on a body to move it over a distance. The work required to lift a body through a height h is given as,

$W=Fh$

Here, (F) is the magnitude of force and (f) is the height.

It can be rewritten as,

$W=mgh$

Here, (m) is the mass of the body.

The tank is in the shape of an inverted right circular cone with height 5 meters and radius 4 meters. As, It is filled with 4 meters of hot chocolate.

Here the ratio of height to radius should be equal to the height at another point (say x), to the radius of that point. Thus,

$\dfrac{5}{4}=\dfrac{x}{r}\\r=\dfrac{4x}{5}$

For the distance x the work done to empty the cone can be given as,

$W=\int^4_0 (5-x)\rho g \pi r^2 dx\\W=\int^4_0 (5-x)1080\times9.81 \pi (\dfrac{4x}{5})^2 dx\\W=22280.4\int^4_0(5-x)x^2 dx\\W=22280.4[\dfrac{128}{3}]\\W=907.96\rm J$

Thus, the work required to empty the tank by pumping the hot chocolate over the top of the tank is 907.96 J.

Learn more about the work done here;

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