Find a basis of the space of solutions in R^n of the equation

$$x_1 + 2x_2 + 3x_3 + \ldots + nx_n = 0.$$

Both situations have the same expected value. Find the missing information. Situation 1: Paul hits a dart target worth 15 points 45% of the time. Situation 2: He hits a dart target worth 10 points x% of the time.

Again using Appendix F, which planet(s) might you expect not to have significant seasonal activity? Why?

Graph each function. y=|3 x|

(a) How is Bottom transformed? (b) Is Bottom aware of his transformation? Explain. (c) Does the transformation alter Bottom's personality as well as his appearance? Why or why not?

A quadratic relation has roots 0 and -6 and a maximum at (-3, 4). Determine the equation of the relation.

Either evaluate the given improper integral or show that it diverges. $\int_{0}^{+\infty} x^{2} e^{-x} d x$

Use the given information about the angle $\theta$ to find the exact value of a. $\sin 2 \theta .$ b. $\cos 2 \theta .$ c. $\tan 2 \theta.$ $\tan \theta = - 2 , \frac { \pi } { 2 } < \theta < \pi$

Find the arc length of the curve over the given interval. $r=6 \cos \theta, 0 \leq \theta \leq 2 \pi.$ Check your answer by geometry.

Mentally calculate each quotient. Write each answer as a decimal number.

$$2.5\div100$$

Simplify each product. Use an area model as needed.

$$2 x ( 3 x + 1 )$$

Metabolic activity in the human body releases approximately $1.0\times10^4\;\mathrm{kJ}$ of heat per day. Assume that a 55-kg body has the same specific heat as water; how much would the body temperature rise if it were an isolated system? How much water must the body eliminate as perspiration to maintain the normal body temperature (98.6 °F)? Comment on your results. (The heat of vaporization of water is 2.41 kJ/g.)

An epidemic hits a city. Each person is classified by the Health Department as either well, sick, or a carrier. The proportion of people in each category by age groups is given by the matrix

$$\begin{array}{lll}{0-15} & {16-35} & {\text { Over } 35} \\ {\text { Sick }} & {0.25} & {0.60} & {0.70} \\ {\text { Sick }} & {0.25} & {0.35} & {0.20} \\ {\text { Carrier }} & {0.10} & {0.05} & {0.10}\end{array} ]=A$$

The population of the city by age and gender is

$$\begin{array}{lll}{0-15} & {35,000} & {30,000} \\ {16-35} & {55,000} & {50,000} \\ {\text { Over } 35} & {70,000} & {75,000}\end{array} ]=B$$

(a) Compute AB. Interpret the meaning of the entries in AB. (b) How many sick males are there? (c) How many females are well?

(A) Identify the type of equation or inequality (some may fit more than one category). (B) Solve the equation or inequality. Write the solution sets to the inequalities in interval notation if possible.

$$(2p+1)(p+5)=2p+40$$