In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

Water leaks through a hole of area $B=0.002 \mathrm{~m}^2$ at the bottom of a cylindrical tank that is filled with water and has height 3 m and a base of area $10 \mathrm{~m}^2$. How long does it take (a) for half of the water to leak out and (b) for the tank to empty ?

The assembly consists of three disks $A, B$, and $C$ that are used to support the load of $140 \mathrm{kN}$. Determine the smallest diameter $d_1$ of the top disk, the diameter $d_2$ within the support space, and the diameter $d_3$ of the hole in the bottom disk. The allowable bearing stress for the material is $\left(\sigma_{\text {allow }}\right)_b=350 \mathrm{MPa}$ and allowable shear stress is $\tau_{\text {allow }}=125 \mathrm{MPa}$.

(a) Consider silicon at T=300 K. Plot the thermal equilibrium electron concentration $n_0$ (on a log scale) over the energy range $0.2 \leq E_c-E_F \leq 0.4 \mathrm{eV}$. (b) Repeat part (a) for the hole concentration over the range $0.2 \leq E_F-E_v \leq 0.4 \mathrm{eV}$.

The Weinreb amide method is a good way to prepare ketones in high yields from acid chlorides.

(a) In the first step of this method, an acid chloride is converted into an amide with $\mathrm{N}, \mathrm{O}$-dimethylhydroxylamine. The product is known generically as a Weinreb amide, after its inventor Steven Weinreb (b. 1941), a Pennsylvania State University chemist. Give the struc-ture of the Weinreb amide by completing the following reaction. (b) The Weinreb amide is allowed to react with a Grignard or organolithium reagent to form a stable tetrahedral addition intermediate. Using a generic structure for the organolithium reagent $\mathrm{R}^{\prime}-\mathrm{Li}$, draw a structure for this intermediate. This intermediate is stable to decomposition for two reasons. What are they? (Hint: The intermediate contains the metal ion from the Grignard or lithium reagent.)

(c) Only after aqueous acid is added does the tetrahedral intermediate break down to the ketone product and other by-products. Write a curved-arrow mechanism for this process and show all the products.

(d) Give the products formed when cyclohexanecarbonyl chloride (A) is converted into a Weinreb amide (B), and this amide is allowed to react with propylmagnesium bromide (C) followed by protonolysis.

(e) Why not simply allow the acid chloride to react with the Grignard or organolithium reagent? Why is the Weinreb amide used instead?

Thee carbonyl becomes negatively charged that is short lived because it is stabilized by H bonding. In the oxyanion hole substrate cleavage an amino leaving group leaves due to the negatively charged carbonyl collapsing back into a double bond displacing the amino group. Which is protonated by His57?

The Figure discussed earlier is a graph of the function

$$h(t)=\frac{t^2-2 t}{(t-2)\left(t^2+1\right)} .$$

The point marked A is an inflection point; point B is a local maximum; and point C is a hole in the graph. Find the coordinates of the point A

Reaves Media Group of Dallas, Texas, has had an excellent year, and the CEO, Henry Reaves, would like to reward "the troops" for their hard work with a rustic yet plush winter retreat. The CEO wants to host a four-day conference/retreat/vacation for his 55 marketing and media professionals with their spouses or significant others at some spectacular winter resort. Ideally, the location would delight any taste, with activities ranging from dining and relaxing in style to downhill and cross-country skiing, snowboarding, snowmobile tours, and other winter sports.

One of the choices is Jackson Hole, Wyoming, a famous ski resort town with steep slopes and dramatic mountain views. As you investigate the options in Jackson Hole, you are captivated by the Four Seasons Resort, a five-star facility with an outdoor pool, a spa tub, ski in/ski out access, and an amply equipped gym and fitness room. Other amenities include an on-site spa with massage and treatment rooms, a sauna, and facial and body treatments. Bathrooms feature separate bathtubs and showers, double sinks, and bathrobes. For business travelers, the hotel offers complimentary wired high-speed Internet access, complimentary wireless Internet access, and multiline phones as well as the use of two desktop computers.

The website of the Four Seasons Jackson Hole is not very explicit on the subject of business and event facilities, so you decide to jot down a few key questions. You estimate that your company will require about 50 rooms. You will also need two conference rooms (to accommodate 25 participants or more) for one and a half days. You want to know about room rates, conference facilities, A/V equipment in the conference rooms, Internet access, and entertainment options for families. You have two periods that would be possible: December 15-19 or January 12-16. You realize that both are peak times, but you wonder whether you can get a discounted group rate. You are interested in entertainment in Jackson Hole, and in tours to the nearby national parks. Jackson Hole airport is 4.5 miles away, and you would like to know whether the hotel operates a shuttle. Also, one evening the CEO will want to host a banquet for about 85 people. Mr. Reaves wants a report from you by September 15.

YOUR TASK. Write a well-organized direct request letter or e-mail to Laura O'Malley, Sales Manager, Four Seasons Resort, 7680 Granite Loop Road, Teton Village, WY 83025.

The Figure discussed earlier is a graph of the function

$$h(t)=\frac{t^2-2 t}{(t-2)\left(t^2+1\right)} .$$

The point marked A is an inflection point; point B is a local maximum; and point C is a hole in the graph. Find the coordinates of the point C

A shaft is made of a steel alloy having an allowable shear stress of $\tau_{\text {allow }}=120 \mathrm{~N} / \mathrm{mm}^2$. If the diameter of the shaft is $15 \mathrm{~mm}$, determine the maximum torque $\mathbf{T}$ that can be transmitted. What would be the maximum torque $\mathbf{T}^{\prime}$ if a $10 \mathrm{~mm}$-diameter hole is bored through the shaft? Sketch the shear-stress distribution along a radial tine in each case.

The Towers of Hanoi is a popular puzzle. It consists of three pegs and a number of discs of differing diameters, each with a hole in the center. The discs initially sit on one of the pegs in order of decreasing diameter (smallest at top, largest at bottom, as in Fig. 5.5), thus forming a triangular tower. The object is to move the tower to one of the other pegs by transferring the discs to any peg one at a time in such a way that no disc is ever placed upon a smaller one.

Give a recurrence relation for $a_n$, the number of moves required to transfer n discs from one peg to another.

Suppose that the conical tank in previous problem is inverted, as shown in figure, and that water leaks out a circular hole of radius 2 in. in the center of the circular base. Is the time it takes to empty a full tank the same as for the tank with vertex down in previous problem? Take the friction/contraction coefficient to be $c=0.6$ and $g=32 \mathrm{ft} / \mathrm{s}^2$.

A one-sided $\mathrm{p}^{+} \mathrm{n}$ silicon diode, which has a forward-bias current of $I_F=1.75 \mathrm{~mA}$, is switched to reverse bias with an effective reverse-biased voltage of $V_R=2 \mathrm{~V}$ and an effective series resistance of $R_R=4 \mathrm{k} \Omega$. The minority carrier hole lifetime is $10^{-7} \mathrm{~s}$. (a) Determine the storage time $t_{s .}$. (b) Calculate the decay time $t_2$. (c) What is the turn-off time of the diode?

The figure shown is a square with a triangular hole cut into one side. The ratio of the height h of the triangle to a side length of the square is 7 to 8 . The ratio of the base b of the triangle to the side length of the square is 1 to 2 . If the area of the square is 64 square inches, what is the area of the shaded part of the square? Show your work.

An ancient device for measuring time is shown . The axisymmetric vessel is shaped so that the water level falls at a constant rate. Determine the shape of the vessel, $R=R(z)$, if the water level is to decrease at a rate of $0.10 \mathrm{~m} / \mathrm{hr}$ and the drain hole is $5.0 \mathrm{~mm}$ in diameter. The device is to operate for $12 \mathrm{hr}$ without needing refilling. Make a scale drawing of the shape of the vessel.