A reservoir has the shape of a truncated cone as shown in the $$gure above. It measures $2$m high. The radius of the base is $1$m and the top radius is $2$m$.$ It is $$lled with water. We want to calculate the amount of work required to pump all of its water through the hose standing $2$m above the reservoir.

$($i$)$ Let x be the height in metres measured from the base of the reservoir. The weight of a thin water layer between heights x and x$+\backslash$Delta x is approximately P$($x$)\backslash$Delta x$.$ What is P$($x$)?$

P$($x$)=$

FORMATTING: Express the answer as a function of x$.$

Recall that the density of water is $\backslash$rho $=1000$Kg$/$m$3$ and the acceleration due to gravity on the earth$$s surface is g$=9\mathrm{.}8$m$/$s$2.$

$($ii$)$ The work in Joules required to pump the thin water layer to $2$m above the reservoir is approximately w$($x$)$ delta x$.$ What is w$($x$)?$

w$($x$)=$

FORMATTING: Express the answer as a formula

$($iii$)$ Using previous results, determine the amount of work required to pump all of the water to $2$m above the reservoir.

FORMATTING: If you round your answer, ensure that the round off error is less than $1\%$ of the value.

Work$=$ Joules