Solve the below triangle:

$$a=9, c=12, \beta=23^{\circ}$$

A sphere of linear dielectric material has embedded in it a uniform free charge density $\rho$. Find the potential at the center of the sphere (relative to infinity), if its radius is R and the dielectric constant is $\epsilon_r$.

Expand the following binomial.

(r + 2q)(r - q)

A gun is fired vertically into a $1.40$-kg block of wood at rest directly above it. If the bullet has a mass of $21.0$ g and a speed of $210$ m/s, how high will the block rise into the air after the bullet becomes embedded in it?

In a dilation every image is ?______ to its preimage.

Expand each binomial. $(11 x-1)^{3}$

Given the adjacency relation $\rho=\{(2,1),(3,2),(3,3),(3,4),(4,5),(6,3),(6,6)\}$ on the set N = {1, 2, 3, 4, 5, 6}, find the corresponding directed graph and adjacency matrix.

Determine the gradients of the following fields: (a) $U=e^{x+2 y} \cosh z$ (b) $T=\frac{3 z}{\rho} \cos \phi$ (c) $W=\frac{5 \cos \phi}{r}+2 r^{2} \sin \phi$

The drag force $F_D$ on a sphere depends on the velocity $V$, the viscosity $\mu$, the density $\rho$, the surface roughness height $e$, the freestream fluctuation intensity $I$ (a dimensionless quantity), and the diameter $D$. Find an expression for $F_D$.

Use the formulas you derived in previous exercises to write formulas for sin (a-b) and sin (a+b) in terms of sine and cosine of a and b. Hint: Use the cofunction identities

$$\sin \left(\frac{\pi}{2}-a\right)=\cos a \text { and } \cos \left(\frac{\pi}{2}-a\right)=\sin a$$

and the fact that

$$\cos \left[\left(\frac{\pi}{2}-a\right)+b\right]=\sin (a-b) \text { and } \sin (a+b)-\sin [a-(-b)] .$$

Simplify Factorial Notation and Binomial Coefficients 9!

Solve what is being asked in the question below.

The function $g(\rho)$ is graphed below. Let $\rho=$ $\sqrt{x^2+y^2+z^2}$. Define $f$, a function of $x, y, z$ by $f(x, y, z)=g\left(\sqrt{x^2+y^2+z^2}\right)$. Let $\vec{F}=\operatorname{grad} f$.

Give a unit vector in the direction of $\vec{F}$ at the point (1, 2, 2).

Expand each binomial. $(3 y-11)^{4}$

Is the transformation $T(a x+b)=a x^2+a x+a$ of $P_1 \rightarrow P_2$ linear? Determine the image of $2 x+1$. Find another element of $P_1$ that has the same image.