MA $122$

Workshop $6$

$12/12/24$

Names $($Printed$)$:

Instructions: Work on the problems below with a small group of other students. Write your solutions on a separate sheet of paper $($you may alternatively use a tablet$),$ and be prepared to turn in your individual work at the end of recitation. Legibility, organization, and clearly stated reasoning are all important!

Ice has a density of approximately $1000k\frac{g}{m^3}.$ How much work is required to build an ice pyramid that is $1$ m high with a square base measuring $2m2m?$ You may assume that $g,$ the acceleration due to gravity, is equal to $10\frac{m}{s^2}.$

The following are conceptual questions related to Problem $1.$ Try to give short and clear explanations.

$($a$)$ If you build a pyramid that is $1$ m tall and has a rectangular base measuring $4m1m,$ will you need the same amount of work as above? $($Think in terms of slices!$)$

$($b$)$ Suppose you dug a hole in the ground that is an inverted pyramid with respect to the one from Problem $1$ and it is filled with water $($also $1000k\frac{g}{m^3}).$ Will the amount of work needed to pump out the water be the same as the amount of work needed to build the pyramid in Problem $1?$ If not, then is it less or more work than in Problem $1?$ Try to answer without any calculations.

A square pyramid$-$shaped tank $($shown in the figure below$)$ is filled with water.

$($a$)$ Give a formula for the approximate volume of a layer of water of thickness $y$ at height $y$ as shown in the figure.

$($b$)$ If the density of water is $1000k\frac{g}{m^3}$ and acceleration due to gravity is $9\mathrm{.}8\frac{m}{s^2},$ find the work required to pump the water to the top of the tank.