$4.)$ A shape is given parametrically as

$($:$rcos(),rsin(),2\sqrt[\phantom{2}]{r}$:$)$

$0r9$

$0<mn>2</mn><mi></mi>$

Find the volume of the shape. Give your answer to $3$ decimal places.

$5.)$ Slime is dripping on the roof of a doghouse, which can be modeled as the top two sides of an equilateral triangular$-$prism pointing up$,$ given by the equation $z=-|x|$ for $-2x2$ and $-4y4.$ The slime is dripping in such a way that, at the considered time, the build$-$up of slime, in lbs per unit area, is given by $S(x,y,z)=2^x+sin(y)+z.$ What is the total weight of slime on the roof?

$6.)$ A pie, initially at temperature $350F,$ is left out to cool. The pie's temperature follows Newton's Law of Cooling and thus the rate of change of the temperature with respect to time is proportional to the difference between the pie's current temperature and the ambient temperature of the room, which is a brisk $65F.$ The constant of proportionality for the pie is $-0\mathrm{.}0823(\frac{1}{\text{h}\text{o}\text{u}\text{r}}).$

a$.$ Make any necessary assumptions.

b$.$ Write down an IVP modeling this situation. Define all variables.

c$.$ What is the temperature of the pie $6$ hours later?