What are the properties that identify a rotation?

Make sense of problems. A sale on dog food that costs $10 a bag says buy 1, get one$$$\frac { 1 } { 2 }$ off. What is the true percent discount on the dog food?

The distance from the surface of a lens to the focal point is the

Write each set in roster form and in set-builder notation. P is the set of natural numbers that are factors of 25.

Abel and Family Perfumes wants to add one or more of four new products to its current line of colognes. Historically, Abel has used a 5-year project recovery period and a MARR of 20% per year. (a) Determine which of the four options the company should undertake on the basis of a present worth analysis, provided the total amount of investment capital available is $800,000. Use a hand solution, unless assigned otherwise. (b) What products are selected if the investment limit is increased to$900,000? Use a spreadsheet, unless assigned otherwise. (All cash flows are in $1000 units.)$ $$ \begin{matrix} \text{Product Line} & \text{R1} & \text{S2} & \text{T3} & \text{U4}\\ \hline \text{Investment, \$} & \text{$-200$} & \text{$-400$} & \text{$-500$} & \text{$-700$}\\ \text{M&O cost, \$/year} & \text{$-50$} & \text{$-200$} & \text{$-300$} & \text{$-400$}\\ \text{Revenue, \$/year} & \text{150} & \text{450} & \text{520} & \text{770}\\ \hline \end{matrix} $$ $

As previously discussed, it suggests that using CFLs is a good idea from a purely financial perspective unless you live in an area where power is relatively inexpensive, but there is another wrinkle. Suppose you have a residence with a lot of incandescent bulbs that are used on average 500 hours a year. The average bulb will be about halfway through its life, so it will have 500 hours remaining (and you can’t tell which bulbs are older or newer). At what cost per kilowatt-hour does it make sense to replace your incandescent bulbs today?

Solve the problem.

The polar equation

$$r=\frac{a\left(1-e^2\right)}{1+e \cos \theta}$$

can be used to graph the orbits of the planets, where $a$ is the average distance in astronomical units from the sun and $e$ is a constant called the eccentricity. The sun will be located at the pole.

Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth's distance from the sun compare to the other planets' distances from the sun?

Find the number of positive outcomes in the sample space. Then list the possible outcomes. You flip a coin and draw a marble at random from a bag containing two purple marbles and one white marble.

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. $34 x^{2}-24 x y+41 y^{2}-25=0$

Determine whether the statement is true or false. Use the following sets of numbers.

$N=$ set of natural numbers.

$Z=$ set of integers

$I=$ set of irrational numbers

$Q=$ set of rational numbers

$\mathbb{R}=$ set of real number

$$Z \subseteq \mathbb{P}$$

Find the complement of the following set.

The set {x|x is an integer strictly between 0 and 8} if U is the set of all integers.

A chiral sample gives a rotation that is close to $180^{\circ}$. How can one tell whether this rotation is $+180^{\circ}$ or $-180^{\circ} ?$

Show that the set of irrational numbers is an uncountable set.

Riley and Rhoda plan to buy several bags of dog food and a dog collar. Each bags of dog food costs $7, and the dog collar costs$5. a. Use the information above to write an expression for the amount of money they will spend if they buy b bags of food. b. Riley and Rhoda have $30. Use this information and the expression you wrote for Part (a) to write an inequality for finding the number of bags of food they could buy. c. Solve the inequality and graph the solutions. Check your answer in the original situation.