A $0\mathrm{.}6-$kg model rocket is loaded with $0\mathrm{.}85$ kg of rocket fuel. After launch, the rocket rises at a constant rate of $4$ m$/$s$,$ but the rocket fuel is dissipated at a rate of $0\mathrm{.}15$ kg$/$s$.$ Find the work done in propelling the rocket $20$ m above the ground. $($Use $9\mathrm{.}8$ m$/$s$2$ for g$.)$

Find the work $($in J$)$ needed to lift the model rocket itself. $($Ignore the work needed to lift the fuel.$)$

J

Show how to approximate the required work $($in J$)$ to lift the rocket fuel by a Riemann sum. $($Let x be the distance in meters above the ground. Enter xi$*$ as xi$.)$

lim$$n$$$$

n

$$xi $=1$

Express the work $($in J$)$ required to lift the rocket fuel as an integral of x $($in m$).$

$0$

$$dx

Evaluate the integral $($in J$).$

J

Find the total work $($in J$)$ by adding the work of lifting the model rocket to the work of lifting the rocket fuel.

J