Approximate a metal needle as a long cylinder with a very pointed, but slightly rounded, end. Sketch the charge distribution and outside electric field lines if the needle has an excess of electrons on it.

Use the figure to determine the solution sets of and

$$2^{3 x+1}>3^{4-x}$$

Use properties of real numbers to write the expression without parentheses.

$$(a-b) 8$$

For the item described in question $8$ , if we expect to sell approximately $15,600$ units next year, how many trips will we need to make to the supplier over the year?

Simplify using properties of exponents.

$\left(-4 z^2\right)^3$

Prove for R a commutative ring with unity, R is a Noetherian ring if and only if every ideal in R has a finite basis.

Identify the properties that are illustrated in each of the following.

$$0+(-7)=-7+0$$

Use a rectangular coordinate to plot

$$\mathbf{u}=\left[\begin{array}{l} 5 \\ 2 \end{array}\right], \mathbf{v}=\left[\begin{array}{r} -2 \\ 4 \end{array}\right]$$

, and their images under the given transformation T. (Make a separate and reasonably large sketch for each exercise.) Describe geometrically what T does to each vector x in $\mathbb{R}^{2}$.

$$T(\mathbf{x})=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$$

A ballet teacher wants to divide his square classroom in half diagonally with masking tape. How much tape will he need if the side length of the classroom is 40 ft?

Your woodshop instructor requires that you cut several pieces of wood within $\frac 3 {16}$ inch of his specifications. Let p represent the specification and let x represent the length of a cut piece of wood. Write an absolute value inequality that describes the acceptable values of x. One piece of wood is specified to be p = 9 $\frac 1 8$ inches. Describe the acceptable lengths for the piece of wood.

Determine the polar equation of a parabola with focus at the pole, satisfying the given condition. The vertical directrix is 4 units to the left of the pole.

What is the intersection between the planes?

Is the graph of $y=\frac{2}{3} x-4$ parallel to the graph of $y=-\frac{3}{2} x-4$?

a. What are some properties of hydrocarbons? b. How are hydrocarbons similar? How are they different?