You are standing at the very rear of a 6.0-m-long, 120-kg raft that is at rest in a lake with its prow only $0.50 \mathrm{~m}$ from the end of the pier (Figure 5-87). Your mass is $60 \mathrm{~kg}$. Neglect frictional forces between the raft and the water.

$(c)$ When you are at the front of the raft, how far are you from the end of the pier?

In the following exercise, verify the Divergence Theorem by evaluating $\int_S \int F \cdot N d S$ as a surface integral and as a triple integral.

$$F(x, y, z) = xzi + zyj + 2z^2k$$

S: surface bounded by $z = 1 − x^2 − y^2$ and z = 0

Find the domain and the range of each relation. Also determine whether the relation is a function.

$$\left\{\left(\frac{3}{2}, \frac{1}{2}\right),\left(1 \frac{1}{2},-7\right) \cdot\left(0, \frac{4}{5}\right)\right\} { }$$

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves $y=e^{-x^2}, \quad y=0, \quad x=0, \quad x=1$ about the $y$-axis.

Suppose A has a right-inverse B. Then AB = I leads to AᵀAB = Aᵀ or B(AᵀA)⁻¹Aᵀ. But that satisfies BA = I; it is a left-inverse. Which step is not justified?

What two probabilities can you multiply in order to find the probability of the event in the previous exercise?

An end milling operation is performed on a $\mathrm{CNC}$ machining center. The total length of travel is $800 \mathrm{~mm}$ along a straight path. Cutting speed is $1.5 \mathrm{~m} / \mathrm{sec}$ and chip load is $0.09 \mathrm{~mm}$. The end mill has two teeth and its diameter $=12.5 \mathrm{~mm}$. Determine (a) feed rate in rev/ min and (b) time to complete the cut.

Underline each word that should be capitalized in the following items. If an item is already correct, write C on the line provided.

Example ______ 1. the $\underline{\text{eiffel}}$ $\underline{\text{tower}}$ observation deck

______the top of the empire state building

Three people are floating on a 120.- $\mathrm{kg}$ raft in the middle of a pond on a warm summer day. They decide to go swimming, and all jump off the raft at the same time and from evenly spaced positions around the perimeter of the raft. One person, with a mass of $62.0 \mathrm{~kg}$, jumps off the raft at a speed of $12.0 \mathrm{~m} / \mathrm{s}$. The second person, of mass $73.0 \mathrm{~kg}$, jumps off the raft at a speed of $8.00 \mathrm{~m} / \mathrm{s}$. The third person, of mass $55.0 \mathrm{~kg}$, jumps off the raft at a speed of $11.0 \mathrm{~m} / \mathrm{s}$. At what speed does the raft drift from its original position?

Explain how to use the Remainder Theorem to evaluate $P(5)$ if $P$ is a polynomial function.

A 60-kg woman stands on the back of a 6.0-m-long, $120-\mathrm{kg}$ raft that is floating at rest in still water. The raft is $0.50 \mathrm{~m}$ from a fixed pier, as shown in Figure 8-52.

$(d)$ On land, the woman puts a lead shot $6.0 \mathrm{~m}$. Then, standing at the back of the raft, she aims forward, and puts the shot so that just after it leaves her hand, it has the same velocity relative to her as it did when she threw it from the ground. Approximately, where does her shot land?

Answer the following questions about "My Two Lives." Academic Vocabulary words are underlined below and defined at the right.

What was a major component of Lahiri's growing identity as an American?

find the inverse function of f. f(x) = x^5 - 2

Estimate sphericities of the following simple particle shapes: (a) a cylindrical needle with a height, H, equal to 5 times the diameter, D; (b) a rectangular prism of sides a, 2a, and 3a.