High-speed elevators function under two limitations: (1) the maximum magnitude of vertical acceleration that a typical human body can experience without discomfort is about $1.2 \mathrm{~m} / \mathrm{s}^2$, and (2) the typical maximum speed attainable is about $9.0 \mathrm{~m} / \mathrm{s}$. You board an elevator on a skyscraper's ground floor and are transported $180 \mathrm{~m}$ above the ground level in three steps: acceleration of magnitude $1.2 \mathrm{~m} / \mathrm{s}^2$ from rest to $9.0 \mathrm{~m} / \mathrm{s}$, followed by constant upward velocity of $9.0 \mathrm{~m} / \mathrm{s}$, then deceleration of magnitude $1.2 \mathrm{~m} / \mathrm{s}^2$ from $9.0 \mathrm{~m} / \mathrm{s}$ to rest. Find the change in the magnitude of the normal force, expressed as a $\%$ of your normal weight during each stage.

For the given problem interpret each of the following row operations as a stage in arriving at the reduced echelon form of a matrix. Why have the indicated operations been selected? What particular aims do they accomplish in terms of the systems of linear equations that are described by the matrices?

(a)

$$\left[\begin{array}{rrrr} 1 & -4 & 3 & 5 \\ -2 & 1 & 7 & 5 \\ 4 & 0 & -3 & 6 \end{array}\right] \quad \begin{gathered} \approx\\ \mathrm{R} 2+(2) \mathrm{R} 1 \\ \mathrm{R} 3+(-4) \mathrm{R} 1 \end{gathered} \quad\left[\begin{array}{rrrr} 1 & -4 & 3 & 5 \\ 0 & -7 & 13 & 15 \\ 0 & 16 & -15 & -14 \end{array}\right]$$

(b)

$$\left[\begin{array}{rrrr}1 & 2 & -4 & 7 \\ 0 & 3 & 9 & -6 \\ 0 & 4 & 7 & -8\end{array}\right] \underset{\left(\frac{1}{3}\right) \mathrm{R} 2}{\approx}\left[\begin{array}{rrrr}1 & 2 & -4 & 7 \\ 0 & 1 & 3 & -2 \\ 0 & 4 & 7 & -8\end{array}\right]$$

(c)

$$\left[\begin{array}{rrrr} 1 & 3 & -4 & 5 \\ 0 & 0 & -2 & 6 \\ 0 & 1 & 3 & -8 \end{array}\right] \quad \begin{gathered} \approx\\ R2\leftrightarrow R3 \end{gathered}\left[\begin{array}{cccr} 1 & 3 & -4 & 5 \\ 0 & 1 & 3 & -8 \\ 0 & 0 & -2 & 6 \end{array}\right]$$

(d)

$$\left[\begin{array}{rrrr} 1 & 2 & 5 & 0 \\ 0 & 1 & 2 & -3 \\ 0 & -3 & 1 & -2 \end{array}\right] \quad \begin{gathered} \approx\\ \mathrm{R} 1+(-2) \mathrm{R} 2 \\ \mathrm{R} 3+(3) \mathrm{R} 2 \end{gathered}\left[\begin{array}{cccr} 1 & 0 & 1 & 6 \\ 0 & 1 & 2 & -3 \\ 0 & 0 & 7 & -11 \end{array}\right]$$

Regarding the two-stage dividend growth model in the chapter, show that the price of a share of stock today can be written as follows:

$$P_0=\frac{D_0 \times\left(1+g_1\right)}{R-g_1} \times\left[1-\left(\frac{1+g_1}{1+R}\right)^{\prime}\right]+\left(\frac{1+g_1}{1+R}\right)^{\prime} \times \frac{D_0 \times\left(1+g_2\right)}{R-g_2}$$

Can you provide an intuitive interpretation of this expression?

Modular exponentiation. Let $b$ be a positive integer. The notation $a^b$ means to multiply $a$ by itself repeated, with a total of $b$ factors of $a$; that is $$ a^b=\underbrace{a\times a\times ....\times a}_{b\text{ times}} $$ The notation for $\mathbb{Z}_n$ is the same. If $a\in\mathbb{Z}_n$ and $b$ is a positive integer, in the context of $\mathbb{Z}_n$ we define $$ a^b=\underbrace{a\otimes a\otimes ...\otimes a}_{b\text{ times}} $$ Please do the following: a. In the context of $\mlathbb{Z}_n$, prove or disprove: $a^b=a^{b\text{ mod }n}$. b. Without the aid of a computer or a calculator, find, in $\mathbb{Z}_{100}$, the value of $3^{64}$. The most horrible way to do this problem is to fully calculate $3^{64}$ and then reduce mod 100 (although this will give the correct answer - why?). A less horrible way is to multiply 3 by itself 64 times, reducing mod 100 at each stage. This requires you to do 63 multiplication problems. Try to do this calculation using only 6 multiplications, including the very first 3$\times$3=9. c. Estimate how many multiplications you need to do to calculate $a^b$ in $\mathbb{Z}_n$. d. Give a sensible definition for $a^0$ in $\mathbb{Z}_n$. e. Give a sensible definition for $a^b$ in $\mathbb{Z}_n$ when $b<0$. Should you be upset that $a^{1}$ already has a meaning?

Revise the following paragraph by correcting the verb tenses.

The original Globe Theatre, built in 1599, (1) will accommodate 3,000 people. This first Globe (2) burn down in 1613. By then, performances of some of Shakespeare's greatest plays (3) take place on the Globe's stage. Today, visitors (4) have enjoyed performances at the new Globe, which opened in 1997. If you visit, you (5) will have found that it holds 1,500 people, far fewer than in Shakespeare's day.

Alfred Hitchcock's 1959 film North by Northwest is a spy thriller of mistaken identity that culminates with a chase seen on Mount Rushmore. In the early stages of production, it was questionable if the movie would even be made because Hitchcock wanted to film the final scenes on the monument and the National Park Service was worried that doing so would "desecrate" the park service's "Shrine of Democracy". At one point, it was suggested that Hitchcock build his own Mount Rushmore in a sound stage in Hollywood. To explore this option, tech crews created a rough scale model that was $0.8 \%$ the size of the actual monument. The faces on Mount Rushmore are 60 feet tall, and made out of granite. Hitchcock's model was made out of concrete.

a. Choose geometric solids that could represent Mount Rushmore. Estimate the amount of concrete in cubic inches that would be needed to create the model. Make assumptions as needed.

b. Use the information from part (a) to compute a reasonable amount of concrete (in cubic yards) needed for Hitchcock's true to life set of Mount Rushmore. Would your answer be an upper or lower bound? Explain.

A $725 \mathrm{~kg}$ two-stage rocket is traveling at a speed of $6.60 \times 10^3 \mathrm{~m} / \mathrm{s}$ away from Earth when a predesigned explosion separates the rocket into two sections of equal mass that then move with a speed of $2.80 \times 10^3 \mathrm{~m} / \mathrm{s}$ relative to each other along the original line of motion. $(a)$ What is the speed and direction of each section (relative to Earth) after the explosion? $(b)$ How much energy was supplied by the explosion? [Hint: What is the change in kinetic energy as a result of the explosion?]

Revise the following paragraph by correcting the verb tenses.

The original Globe Theatre, built in 1599, (1) will accommodate 3,000 people. This first Globe (2) burn down in 1613. By then, performances of some of Shakespeare's greatest plays (3) take place on the Globe's stage. Today, visitors (4) have enjoyed performances at the new Globe, which opened in 1997. If you visit, you (5) will have found that it holds 1,500 people, far fewer than in Shakespeare's day.

(1) At age ten, I attended my first live theater performance. (2) I followed my mom into the theater. (3) I agreed to watch the performance even though it did not interest me. (4) As we took our seats, I noticed no one was eating popcorn or candy. (5) This was not the movies or even TV! (6) The houselights dimmed. (7) The stage lights brightened at the same time. (8) Everyone became perfectly silent. (9) I was astonished as the characters began to speak. (10) I could have reached out and touched them. (11) I could see and almost feel the texture of their clothing, smell the faintly dusty odor of the stage. (12) The audience clapped at the end of each act. (13) I could feel the sadness ripple through the audience when the hero died! (14) Applauding in unison, the play ended, everyone stood. (15) I felt exhilarated, my smile a mile wide. (16) It was a unique experience. (17) To this day, I still love live theater.

Which sentence contains an absolute phrase?

A. sentence 10

B. sentence 11

C. sentence 13

D. sentence 15

The leaf consists of the petiole and the blade.

Revise the following paragraph by correcting the verb tenses.

The original Globe Theatre, built in 1599, (1) will accommodate 3,000 people. This first Globe (2) burn down in 1613. By then, performances of some of Shakespeare's greatest plays (3) take place on the Globe's stage. Today, visitors (4) have enjoyed performances at the new Globe, which opened in 1997. If you visit, you (5) will have found that it holds 1,500 people, far fewer than in Shakespeare's day.

(1) At age ten, I attended my first live theater performance. (2) I followed my mom into the theater. (3) I agreed to watch the performance even though it did not interest me. (4) As we took our seats, I noticed no one was eating popcorn or candy. (5) This was not the movies or even TV! (6) The houselights dimmed. (7) The stage lights brightened at the same time. (8) Everyone became perfectly silent. (9) I was astonished as the characters began to speak. (10) I could have reached out and touched them. (11) I could see and almost feel the texture of their clothing, smell the faintly dusty odor of the stage. (12) The audience clapped at the end of each act. (13) I could feel the sadness ripple through the audience when the hero died! (14) Applauding in unison, the play ended, everyone stood. (15) I felt exhilarated, my smile a mile wide. (16) It was a unique experience. (17) To this day, I still love live theater.

Which revision to sentence 14 includes an absolute phrase?

A. As the play ended, everyone stood, applauding in unison.

B. The play ended, everyone standing and applauding in unison.

C. The play ended with people applauding in unison, and everyone stood.

D. Standing and applauding in unison at the end, the people really enjoyed the play.

This problem is based on an account by the historian Flavius Josephus, who was part of a band of 41 Jewish rebels trapped in a cave by the Romans during the Jewish Roman war of the first century. The rebels preferred suicide to capture; they decided to form a circle and to repeatedly count off around the circle, killing every third rebel left alive. However, Josephus and another rebel did not want to be killed this way; they determined the positions where they should stand to be the last two rebels remaining alive. The variation we consider begins with n people, numbered 1 to n, standing around a circle. In each stage, every second person still left alive is eliminated until only one survives. We denote the number of the survivor by J (n). Determine J (100), J (1000), and J (10,000) from your formula for J (n).

What is a leaf blade? ____________________