A container in the shape of an inverted pyramid with a...

A container in the shape of an inverted pyramid with a square base of $2 \mathrm{~m}$ by $2 \mathrm{~m}$ and height of $5 \mathrm{~m}$ is filled to a depth of $4 \mathrm{~m}$ with corn slurry. How much work is required to pump all the slurry over the top of the container? (The density of corn slurry is $\left.\rho=17.9 \mathrm{~kg} / \mathrm{m}^{3} .\right)$

Key Concepts

Work

Work is the energy required to move an object over a distance. In fluid pumping problems, work is computed for each infinitesimally small volume of fluid that must be moved, and the contributions are summed via integration. This typically involves calculating the gravitational force (weight) acting on the fluid and the vertical distance over which each volume element is lifted.

Calculus Integration

Integration is used to sum the contributions of energy required to move each small slice of fluid. By setting up a definite integral over the depth of the fluid, where the integrand represents the product of weight and lifting distance for an infinitesimal slice, one can determine the total work required.

Similar Triangles

When dealing with containers that have a shape where cross-sectional area varies with height, the use of similar triangles helps relate the dimensions of a horizontal slice at any given depth to the overall dimensions of the container. This geometric relationship is crucial for expressing the area of the slice as a function of its vertical position.

Density and Weight

Density allows conversion of a fluid's volume into its mass, and multiplying this mass by the acceleration due to gravity gives the weight. This is essential for calculating the force involved in lifting each infinitesimal slice of the fluid during the pumping process.

Inverted Pyramid Geometry

The geometry of an inverted pyramid defines how its cross-sectional area changes with height. Understanding this changing geometry is key in establishing the function that describes the area (and therefore the volume) of horizontal slices at different vertical positions, which directly affects the calculation of work through integration.

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