# Math Help - Conical paper cup - Work problem

1. ## Conical paper cup - Work problem

A large conical paper cup has height 1 m. and top radius $\frac{1}{4} m.$ It is filled with water to a height of $\frac{3}{4} m$. Write (but do not evaluate) an integral that equals the work required to empty the cup by pumping all of the water to the top. (Note: The density of water is $1000kg/m^3$. Answer needs to be in terms of a single variable.

**I'm sure there is a good explanation in a previous thread that I could use. If someone knows of one and can direct me there, that's fine. Or if someone feels like walking me through this, I'd love it! I never really fully understood these work problems.
Thanks

2. Originally Posted by mollymcf2009
A large conical paper cup has height 1 m. and top radius $\frac{1}{4} m.$ It is filled with water to a height of $\frac{3}{4} m$. Write (but do not evaluate) an integral that equals the work required to empty the cup by pumping all of the water to the top. (Note: The density of water is $1000kg/m^3$. Answer needs to be in terms of a single variable.
work = $\int WALT$

$W$ = weight density
$A$ = cross-sectional area of a representative horizontal slice of liquid in terms of y
$L$ = lift distance of a horizontal slice in terms of y
$T$ = slice thickness ... $dy$

$W = \frac{mg}{V} = 9800 \, \frac{N}{m^3}$

$A = \pi \left(\frac{y}{4}\right)^2$

$L = 1 - y$

$T = dy$

work in N-m ...

$\int_0^{\frac{3}{4}} 9800 \cdot \pi \left(\frac{y}{4}\right)^2 \cdot (1-y) \, dy$