(Down)

Find the LCM.

$$8 x , 25 y$$

An environmentalist group collects a liter of water from each of 45 randomly chosen locations along a steam and measures the amount of dissolved oxygen in each specimen. The mean is 4.62 milligrams per liter (mg/L). Is this strong evidence that the stream has a mean dissolved oxygen content of less than 5 mg per liter?

Marley needs to construct a quadrilateral with side lengths of 2, 3, 4, and 5 centimeters. She thinks that there is only one possible quadrilateral that she can make. Is she correct?

You cut a square hole

$$\frac { 1 } { 4 }$$

inch wide in a piece of cardboard. With the cardboard 30 inches from your face the moon fits exactly into the square hole. If the moon is about 240,000 miles from Earth, estimate the moon's diameter.

In one year, there were 116 homicide deaths in Richmond, Virginia. For a randomly selected day, find the probability that the number of homicide deaths is 1. Compare the calculated probabilities to these actual results: 268 days (no homicides); 79 days (1 homicide); 17 days (2 homicides); 1 day (3 homicides); no days with more than 3 homicides.

Divide. $\left(2 x^{3}+9 x^{2}+14 x+5\right) \div(2 x+1)$

Let $W$ be a subspace of the inner product space $V$. Prove that the set

$$W^\perp = \{v \in V: \langle v, w\rangle = 0 for\ all\ w \in W \}$$

is a subspace of $V$.

Getting Started: To prove that $W^\perp$ is a subspace of $V,$ you must show that $W^\perp$ is nonempty and that the closure conditions for a subspace hold.

$(i)$ Find a vector in $W^\perp$ to conclude that it is nonempty.

$(ii)$ To show the closure of $W^\perp$ under addition, you need to show that $\langle v_1 + v_2, w \rangle = 0 for\ all\ w \in W$ and for any $v_1, v_2 \in W^\perp.$ Use the properties of inner products and the fact that $\langle v_1, w \rangle$ and $\langle v_2, w \rangle$ are both zero to show this.

$(iii)$ To show closure under multiplication by a scalar, proceed as in part $(ii)$. Use the properties of inner products and the condition of belonging to $W^\perp$.

Use Cramer’s rule to solve the system of equation.

$$\left\{ \begin{array} { l } { 3 x - 4 y = 9 } \\ { x + 2 y = 8 } \end{array} \right.$$

What is a geometric series? Give some examples of convergent and divergent geometric series.

The perimeter of a triangle is at most 32 cm. One side is 11 cm long. The other two sides are the same length,Draw a diagram and show your work.

Arrange these fractions from least to greatest.

$$\frac { 2 } { 4 } , \frac { 2 } { 5 } , \frac { 2 } { 3 }$$

Which of the series converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

$$\sum _ { n = 3 } ^ { \infty } \frac { ( 1 / n ) } { ( \ln n ) \sqrt { \ln ^ { 2 } n - 1 } }$$

What economic, social, and political crisis did Europe experience in the first half of the seventeenth century?