Problem $4.(1$ point$)$

A rocket accelerates by burning its on$-$board fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff $($including its fuel$)$ is $m,$ the fuel is consumed at rate $r,$ and the exhaust gases are ejected with constant velocity $v_c($relative to the rocket$).$ A model for the velocity of the rocket at time $t$ is given by the equation

$v(t)=-gt-v_{c}ln(\frac{m-rt}{m})$

where $g$ is the acceleration due to gravity and $t$ is not too large. If $g=9\mathrm{.}8\frac{m}{s^2},m=35000kg,r=160k\frac{g}{s},$ and $v_c=3000\frac{m}{s},$ find the height $($in meters$)$ of the rocket one minute after liftoff.

Height $=$ meters

Pro ${?}^3$

$6.(1$ point$)$

ow large the number a has to be so that