(a) What is the ratio of the gravitational force that Earth exerts on the Sun in the winter and the force that exerts in the summer? (b) How many correct answers can you give for part (a)? (Hint: Earth's orbit is an ellipse with the Sun located at one of the foci of the ellipse. Think about the location of the observer on Earth.)

Choose the correct pronoun from each pair in parentheses and draw a line under it.

How surprised we were when the last person to come to the party was (she, her).

Write each percent as a fraction in simplest form. 225%

Write the fraction in simplest form. $\frac{30}{48}$

What gravitational force does each exert on the other when considering the Sun-Earth system?

Consider that the mass of the Earth is $M_E=6\times 10^{24}\ {\rm{kg}}$, for the Sun we have $M_S=2\times 10^{30}\ {\rm{kg}}$, and they are separated approximately by $150\times 10^{6}\ {\rm{km}}$.

Why are fimbriae necessary for colonization?

Evaluate the expression and write in the form $a+b i$.

(a) $\frac{4+2 i}{2-i}$

(b) $(1-\sqrt{-1})(1+\sqrt{-1})$

At time t=0, you begin counting the arrivals of buses at a depot. The number of buses $K_{i}$ that arrive between time i-1 minutes and time i minutes has the Poisson PMF

$$P_{K_{i}}(k)=\left\{\begin{array}{ll}{2^{k} e^{-2} / k !} & {k=0,1,2, \ldots} \\ {0} & {\mathrm { otherwise. }}\end{array}\right.$$

$K_{1}, K_{2}, \dots$ are an iid random sequence. Let $R_{i}=K_{1}+K_{2}+\cdots+K_{i}$ denote the number of buses arriving in the first i minutes. (a) What is the moment generating function $\phi_{K_{i}}(s)$? (b) Find the MGF $\phi_{R_{i}}(s)$. (c) Find the PMF $P_{R_{i}}(r)$. Hint: Compare $\phi_{R_{i}}(s)$ and $\phi_{K_{i}}(s)$. (d) Find $\mathrm{E}\left[R_{i}\right]$ and $\operatorname{Var}\left[R_{i}\right]$.

Calculate.

$$[(-2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}) \times \mathbf{i}] \times[\mathbf{i}+\mathbf{j}]$$

$$\text { Suppose that } a \text { and } b \text { belong to an integral domain, } b \neq 0 , \text { and } a \text { is }\text { not a unit. Show that } \langle a b \rangle \text { is a proper subset of } \langle b \rangle$$

Tell whether you would add a subject (S), a verb $(V)$, or a main clause ( $M$ ) to form a complete sentence.

Tried to warn Tisha about taking on too many activities.

Refer to the formula $F=\frac{G m_1 m_2}{d^2}$. This gives the gravitational force $F$ (in Newtons, N) between two masses $m_1$ and $m_2$ (each measured in $\mathrm{kg}$ ) that are a distance of $d$ meters apart. In the formula, $G=6.6726 \times 10^{-11} \mathrm{~N}^2 / \mathrm{mg}^2$.

Determine the gravitational force between the Earth (mass $=5.98 \times 10^{24} \mathrm{~kg}$ ) and an $80 \mathrm{kg}$ human standing at sea level. The mean radius of the Earth is approximately $6371 \times 10^{\circ} \mathrm{m}$.

In each of the following sentences, cross out any incorrect word and write the correct word above it. If all the words in a sentence are correct, write C after the sentence.

Example: The tounsts had an appomtment to meet with their nation's $\overset{\text{consul}}{\cancel{\text{counsel}}}$.

There sure they're first-aid course w ill be quite useful.

The International Space Station is in orbit around the Earth at a distance $r$ from the center of Earth. A recent addition increased the Station's mass by a factor of 3. Did Earth's gravitational force on the Station increase, decrease, or stay the same and by how much?