Consider the hemispherical closed surface in the figure below. The hemisphere is in a uniform magnetic field that makes an angle ? with the vertical.

(a) Calculate the magnetic flux (?B) through the flat surface S1. (Use any variable or symbol stated above along with the following as necessary: pie.)

?B = __________.

(b) Calculate the magnetic flux (?B) through the hemispherical surface S2. (Use any variable or symbol stated above along with the following as necessary: pie.)

?B =__________.

Assume that the DJIA closed at 11,875 one day recently, and the divisor was 0.12493117.

a. What is the sum of the prices of the 20 stocks in the index, given this information?

b. Assume that one stock in the index, Pfizer, moved $4.40 that day, while the index itself moved about 105 points (to close at 10,875). What percentage of the total movement in the DJIA that day was accounted for by the movement in Pfizer?

c. Now assume that one of the 20 stocks had an initial price of $47.50 and the same stock had a 4-for-1 stock split that day. What would the new divisor have to be to keep the index unchanged at 11,875?

Suppose that the closed economy of an island H is described by the following equations: GDP (Y) = 10000, government expenditures (G) = 600, Taxes (T) = 2000, Consumption (C) = 400 + 3/4 (Y-T), and investment (I) = 200 – 1400 r

a. Write and explain the equation of the GDP

b. Compute

1. Private saving

2. Public saving

3. National saving

4. Equilibrium interest rate

c. What can you conclude about the economy of Island H?

Suppose f maps a closed and bounded set D to the reals is continuous. Then the Uniform Continuity Theorem says f is uniformly continuous on D. To prove this, we will suppose it is not true, and arrive at a contradiction.

So, suppose f is not uniformly continuous on D. If f is not uniformly continuous on D, then there exists epsilon greater than zero and sequences (a_n) and (b_n) in D for which |a_n -b_n| < 1/n and |f(a_n) - f(b_n)| > epsilon. Give comment as to why this must be true if f is not uniformly continuous on D.

Nothing says the sequences (a_n) and (b_n) converge. But, because f is continuous on D, there exists subsequences of (a_n) and (b_n), say (a_nk) and (b_nk) respectively, that do converge. Give comment as to why this would be true.

Show that both limits of the subsequences must lie in D and that they are in fact equal. Explain why this is leads to a contradiction. Finish the proof.

A 0.75 L bottle is cleaned, dried, and closed in a room where the air is 20°C and 44% relative humidity (that is, the water vapor in the air is 0.44 of the equilibrium vapor pressure at 20°C). The bottle is brought outside and stored at 0.0°C. (See Table 5.2) vapor pressure of 20 degrees is 17.5 torr and 0 degrees is 4.6 torr

What mass of water condenses inside the bottle?

A. Find the length of an organ pipe closed at one end that produces a first overtone frequency of 276 Hz when air temperature is 21.3ºC .

B. Find the length of an organ pipe closed at one end that produces a first overtone frequency of 276 Hz when air temperature is 21.3ºC .

C. Electromagnetic radiation having a 73.8 μm wavelength is classified as infrared radiation. What is its frequency in 10¹³ Hz?

D. Find the intensity of an electromagnetic wave having a peak magnetic field strength of 6.04×10⁻⁹ T? Give your answer in milliwatts per square meter.

E.Calculate the ratio of the highest to lowest frequencies of electromagnetic waves that an electronic sensor can see, given the wavelength range of the sensor is from 466 to 600 nm.

Water in a closed jar that contains 0.05000 mg/ L of bicarbonate, as the ion, at a pH of 8.250. Find:

a) Normality of all acid-base species (eq/L) 1 pt.

b) Exact Alkalinity of the solution (Use four significant figures; mg/L as CaCO₃) at 25°C. 1 pt.

Hint: Bicarbonate is not the only carbonate species in the water. Use the equilib. relationship:

HCO₃^- Û H⁺ + CO₃^2-   pK_a2 = 10.33 at 25°C [Table 2-3 EE&S]

c) Approximate Alkalinity of the solution by assuming that all alkalinity is due to bicarbonate and hydroxyl. Is the approximation good? Would it be valid at pH = 12? Why? 1 pt

A closed rectangular box of volume 324 cubic inches is to be made with a square base. If the material for the bottom costs twice per square inch as much as the material for the sides and top, find the dimensions of the box that minimize the cost of materials.

Given a sequence of closed intervals arranged so that each interval is a subinterval of the one preceding it and so that the lengths of the intervals shrink to zero, then there is exactly one point that belongs to every interval of the sequence. (This is known as the nested interval property)