A soap bubble will enclose the maximum space with a minimum amount of surface material. Architects have used this principle to create buildings that enclose a great amount of space with a small amount of building material. If a soap bubble has a surface area of A, then its volume V is given by the equation $V=0.094 \sqrt{A^{3}}$. Find the surface area of a bubble with a volume of $1.75 \times 10^{2}$ cubic millimetres.

What is the best way to paraphrase the sentence below (line 35)? "The principle of truth may itself be carried into an absurdity." A. Being honest is ridiculous. B. Too much candor makes one ludicrous. C. People who follow principles are often persecuted with scorn. D. Carrying one's emotions around openly can lead to ridicule. E. Lying is a normal, everyday practice.

Suppose you attempt to pick up a very heavy object. Before you tried to pick it up, the object was sitting still—its momentum was not changing. You pull very hard, but do not succeed in moving the object. Is this a violation of the Momentum Principle? How can you be exerting a large force on the object without causing a change in its momentum? What does change when you apply this force?

Show that, in principle, $\mathrm{Na}_2 \mathrm{CO}_3(\mathrm{aq})$ can be converted almost completely to $\mathrm{NaOH}(\mathrm{aq})$ by the reaction

$$\begin{aligned} \mathrm{Ca}(\mathrm{OH})_2(\mathrm{~s})+\mathrm{Na}_2 \mathrm{CO}_3(\mathrm{aq}) & \longrightarrow \\ & \mathrm{CaCO}_3(\mathrm{~s})+2 \mathrm{NaOH}(\mathrm{aq}) \end{aligned}$$

Solve each minimum problem using the duality principle. Minimize

$$C = 2x_1 +3x_2+4x_3$$

subject to the constraints 2

$$\left\{ \begin{array}{rcl} x_1 & -2x_2 & -3x_3 \geq -2 \\ x_1 & +x_2 & +x_3 \geq 2 \\ 2x_1 & & +x_3 \geq 3 \\ x_1 \geq 0, & x_2 \geq 0, & x_3\geq 0 \end{array} \right.$$

use Archimedes’ Principle, which states that when a sphere of radius r with density

$$d_s$$

is placed in a liquid of density

$$d_L=62.5lb/ft^3$$

,it will sink to a depth h where .

$$\dfrac{π}{3}(3rh^2-h^3)d_L=\dfrac{4}{3}πr^3d_s$$

Find an approximate value for h if:

$$r=5 ft$$

and

$$d_s=20lb/ft^3$$

Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of vx = 100 m/s. The rifle’s barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Underline any verbs that are used incorrectly. In the space provided, write the correct form (or $C$ if correct) and the number of the $\mathrm{G} / \mathrm{M}$ principle illustrated. When you finish, compare your responses with those provided at the bottom of this page. If your responses differ, study carefully the principles in parentheses.

The reports have laid on his desk for 11 days and are now overdue.

The all-or-none principle states that

A) only motor stimuli can activate action potentials.

B) only sensory stimuli can activate action potentials.

C) all stimuli great enough to bring the membrane to the threshold will produce action potentials of identical magnitude.

D) the greater the magnitude of the stimuli, the greater the intensity of the action potential.

E) all stimuli will produce identical action potentials.

An electron in an infinite square well with $L=10^{-12} \mathrm{~m}$ is moving at relativistic speed; hence, the momentum is not given by $p=(2 m E)^{1 / 2}$. (a) Use the uncertainty principle to verify that the speed is relativistic. $(b)$ Derive an expression for the electron's allowed energy levels and $(c)$ compute $E_1$. (d) By what fraction does $E_1$ computed in (c) differ from the nonrelativistic $E_1$ ?

Use the counting principle to determine the answer to part (a). Assume that each event is equally likely to occur. Two coins are tossed. (a) Determine the number of sample points in the sample space. (b) Construct a tree diagram and list the sample space. Determine the probability that (c) No tails are tossed. (d) Exactly one tail is tossed. (e) Two tails are tossed. (f) At least one tail is tossed.

Estimate the binding energy of the $\mathrm{H}_{2}$ molecule by calculating the difference in kinetic energy of the electrons between when they are in separate atoms and when they are in the molecule, using the uncertainty principle. Take $\Delta x$ for the electrons in the separated atoms to be the radius of the first Bohr orbit, 0.053 nm, and for the molecule take $\Delta x$ to be the separation of the nuclei, 0.074 nm. [Hint let $p \approx \Delta p_{x} .$]

Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of $v_{x}=100 \mathrm{m} / \mathrm{s} .$ The rifle's barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Hoot Enterprises buys a warehouse for $590,000 to use for its East Coast distribution operations. On the date of the purchase, a professional appraisal shows a value of$650,000 for the warehouse. The seller had originally purchased the building for $480,000. Hoot has a similar warehouse on the West Coast that has a book value of$603,000. Under the historical cost principle, Hoot should record the building for

a. $650,000.

b.$480,000.

c. $590,000.

d.$603,000.