If designed properly, a good ammeter should have a very small resistance. Why? Explain clearly, using Kirchhoff's laws.

Verify the Divergence Theorem by evaluating $\int_{S} \int \mathbf{F} \cdot \mathbf{N} d \mathbf{S}$ as a surface integral and as a triple integral. $\mathbf{F}(x, y, z)=2 x \mathbf{i}-2 y \mathbf{j}+z^{2} \mathbf{k}$. S: cylinder $x^{2}+y^{2}=4$, $0 \leq z \leq 3$

A matrix A is called symmetric if $A^{T}=A.$ Verify, for all $3 \times 3$ matrices B, that $B+B^{T}$ is symmetric.

What is absolute advantage? What is comparative advantage?

Show that the largest rectangular box having a fixed surface area must be a cube.

Find each limit using properties of limits. $\lim _{t \rightarrow 2}(t \sqrt{2 t}+4)^{1 / 3}$

Find each limit using properties of limits. $\lim _{t \rightarrow 2} \sqrt{3 t+4}$

Calculate the Hessian matrix Hf(a) for the indicated function f at the indicated point a. $f(x, y, z)=x^{3}+x^{2} y-y z^{2}+2 z^{3}, \mathbf{a}=(1,0,1)$

In Japan, one worker can make 5 tons of rubber or 80 radios. In Malaysia, one worker can make 10 tons of rubber or 40 radios. a. Who has the absolute advantage in the production of rubber or radios? How can you tell? b. Calculate the opportunity cost of producing 80 additional radios in Japan and in Malaysia. (Your calculation may involve fractions, which is fine.) Which country has a comparative advantage in the production of radios? c. Calculate the opportunity cost of producing 10 additional tons of rubber in Japan and in Malaysia. Which country has a comparative advantage in producing rubber? d. In this example, does each country have an absolute advantage and a comparative advantage in the same good? e. In what product should Japan specialize? In what product should Malaysia specialize?

A demand function is p=400-2q, where q is the quantity of the good sold for price $p. Differentiate R with respect to q to find the marginal revenue, MR, in terms of q. Calculate the marginal revenue when q=10.

A 200-L rigid tank contains propane at 400 K, 3.5 MPa. A valve is opened, and propane flows out until half of the initial mass has escaped, at which point the valve is closed. During this process, the mass remaining inside the tank expands according to the relation Pv1.4 = constant. Calculate the heat transfer to the tank during the process.

What size facility should be built so the probability of the patient load’s exceeding the clinic capacity is .10? .30? REFERENCE: 4.80 : Federal resources have been tentatively approved for the construction of an outpatient clinic. In order to design a facility that will handle patient load requirements and stay within a limited budget, the designers studied patient demand. From studying a similar facility in the area, they found that the distribution of the number of patients requiring hospitalization during a week could be approximated by a normal distribution with a mean of 125 and a standard deviation of 32. a. Use the Empirical Rule to describe the distribution of y, the number of patients requesting service in a week. b. If the facility was built with a 160-patient capacity, what fraction of the weeks might the clinic be unable to handle the demand?

Blast furnace gas in a steel mill is available at 500 F to be burned for the generation of steam. The composition of this gas is as follows on a volumetric basis: Component CH4H2COCO2N2H2O Percent by volume 0.12.423.314.456.43.4 Find the lower heating value (Btu/ft3) of this gas at 500 F and P0.

Water at 10$^{\circ} \mathrm{C}$ flows over a flat plate at a speed of 0.8 m/s. The plate is 0.35 m long and 1 m wide. The boundary layer on each surface of the plate is laminar. Assume that the velocity profile may be approximated as linear. Determine the drag force on the plate.