In each case, express the given vector field V in the standard form $\sum v_{i} U_{i}.$ (a) $2 z^{2} U_{1}=7 V+x y U_{3}$ (b) $V(\mathbf{p})=\left(p_{1}, p_{3}-p_{1}, 0\right)_{p}$ for all $\mathbf{p}.$ (c) $V=2\left(x U_{1}+y U_{2}\right)-x\left(U_{1}-y^{2} U_{3}\right)$ (d) At each point $\mathbf{p}, V(\mathbf{p})$ is the vector from the point $\left(p_{1}, p_{2}, p_{3}\right)$ to the point $\left(1+p_{1}, p_{2} p_{3}, p_{2}\right)$ (e) At each point $\mathbf{p}, V(\mathbf{p})$ is the vector from $\mathbf{p}$ to the origin.

An enzyme is a protein that acts as a catalyst for increasing the rate of a chemical reaction that occurs in cells. In a certain reaction, an enzyme A is converted to another enzyme, B. Enzyme B acts as a catalyst for its own formation. Let p be the amount of enzyme B at time t and I be the total amount of both enzymes when t=0. Suppose the rate of formation of B is proportional to p(I-p). Without directly using calculus, find the value of p for which the rate of formation will be a maximum.

Find the real solution(s) of the equation involving absolute value. Check your solution(s). |x+1|=2

The table shows the percent y (in decimal form) of battery power remaining x hours after you turn on a laptop computer.

$$\begin{matrix} \text{Hours, x} & \text{0} & \text{2} & \text{4}\\ \text{Power Remaining, y} & \text{1.0} & \text{0.6} & \text{0.2}\\ \end{matrix}$$

a. Write and graph a linear function that relates y to x. b. Interpret the slope, the x-intercept, and the y-intercept. c. After how many hours is the battery power at 75%?

Add commas where necessary. Delete commas used incorrectly using the delete symbol. Unlike most owls the small saw-whet owl is quite tame.

Evaluate each expression for r = -3 and s = 5. $2^{-4} r^{3} s^{-2}$

The table shows the length y (in inches) of a person's hair after x months.

$$\begin{matrix} \text{Months, x} & \text{Hair Length, y}\\ \text{0} & \text{11.0}\\ \text{3} & \text{12.5}\\ \text{6} & \text{14.0}\\ \end{matrix}$$

a. Write and graph a linear function that relates y to x. b. Interpret the slope and the y-intercept.

A charitable organization is having a raffle for a single prize of $7000. Each raffle ticket costs$3, and 9000 tickets have been sold. (a) Find the expected gain for the purchaser of a single ticket. (b) Find the expected gain for the purchaser of two tickets.

You randomly survey students in your school about the color of their eyes. The results are shown in the tables.

$$\begin{matrix} \text{Eye Color of Males Surveyed} & \text{ } & \text{ }\\ \text{Green} & \text{Blue} & \text{Brown}\\ \text{5} & \text{16} & \text{27}\\ \text{ }\\ \text{ }\\ \end{matrix}$$

$$\begin{matrix} \text{Eye Color of Females Surveyed} & \text{ } & \text{ }\\ \text{Green} & \text{Blue} & \text{Brown}\\ \text{3} & \text{19} & \text{18}\\ \text{ }\\ \text{ }\\ \end{matrix}$$

a. Make a two-way table. b. Find and interpret the marginal frequencies for the survey. c. For each eye color, what percent of the students in the survey are male? female? Organize the results in a two-way table.

A 5.0 kg bucket of water is raised from a well by a rope. If the upward acceleration of the bucket is $3.0 \mathrm{m} / \mathrm{s}^{2}$, find the force exerted by the rope on the bucket of water.

Determine by substitution which of the given numbers, if any, satisfy the given equation. $10-7 x=-x^{2} ; 2,4$

Kevin's office cubicle measures 8 ft by 12 ft. Write an equation to find the amount x that should be added to the current length and width to double the area.

Which compound forms the higher concentration of hydrate at equilibrium, PhCOCHO or $\mathrm{phCH}_{2} \mathrm{CHO}$? Explain your reasoning

Add commas where necessary. Cross out incorrect commas using the delete symbol. On Saturday morning I need to mow the lawn, pull the weeds and sweep the front porch.