Use the Divergence Theorem to calculate the surface integral int int_S F · dS, that is, calculate the flux of F across S. F(x, y, z) = (x^3 + y^3)i + (y^3 + z^3)j + (z^3 + x^3)k, S is the sphere with center the origin and radius 2.

The half-life of C-14 is about 5730 yr.
a) Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only ${\displaystyle 77\mathrm{\%}}$ of the C-14 originally in the dye remains. When was the cloth painted?
b) A well-preserved piece of wood found at an archaeological site has 6.2% of the C-14 that it had when it was alive. Estimate when the wood was cut.

Let ${\displaystyle X_1,X_2,\dot{s},X_n}$ be n independent random variables each with mean 100 and standard deviation 30. Let X be the sum of these random variables. Find n such that Pr${\displaystyle \left(X>2000\right)\ge 0.95}$

If z and ${\displaystyle \bar{z}}$ are conjugate complex numbers , find two complex numbers, ${\displaystyle z=z_1\text{ and }z=z_2}$ , that satisfy the equation ${\displaystyle 3z\bar{z}+2(z-\bar{z})=39+j12}$

Write a multivariable function (i.e z=f(x,y)) for a linear function that contains the points (−1,0),(0,2),(1,−1). Ive

Average value over a multivariable function using triple integrals. Find the average value of ${\displaystyle F(x,y,z)=x^2+y^2+z^2}$ over the cube in the first octant bounded bt the coordinate planes and the planes $x=5$, $y=5$, and $z=5$

Equilibrium Price: Skateboards
The demand for your factory-made skateboards, in weekly sales, is
${\displaystyle q=-7p+50}$
If the selling price is $p. If you are selling them at that price, you can obtain
${\displaystyle q=3p-30}$
Per week from the factory. At what price should you sell your skateboards so that there is neither a shortage nor a surplus? (Round your answer to the nearest cent.)

Use Green's Theorem to evaluate ${\displaystyle {\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}}$ where ${\displaystyle \overrightarrow{F}(x,y)=x{y}^{2}i+(1-x{y}^3)j}$ and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4).
The orientation of C is counterclockwise.