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

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

$q,$

$117\mathrm{.}6$

$$

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 $x_i^{**}$ as $x_i.)$

$\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}(\frac{4\mathrm{.}9*20}{n})x$

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

${\displaystyle \underset{0}{\overset{20}{}}}(4\mathrm{.}9)x$

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

$94\mathrm{.}08J$

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

$219\mathrm{.}52J$