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.

2. A circuit contains 3 resistors connected in series and no battery. It is a closed circuit and has dimensions of 18 cm by 42 cm. The resistor values are 12, 15 and 18 ohms. A magnetic field directed out of the plane of the circuit changes from 52 T to 0 T in 2.9 sec.

a. Sketch the circuit and indicate the direction of the initial magnetic field.

b. What is the magnetic flux through the circuit when the magnetic field is at its starting value? What is the flux when the field is at its final value?

c. What is the direction of the current in the circuit as the magnetic field is changing? Explain.

d. What is the induced emf and current in the circuit as the magnetic field is changing?

e. What is the direction of the current after the magnetic field reaches its final value?

a dead body was found within a closed room of a house where the temperature was at 21°C. at the time of discovery the core temperature of the body was determined to be 28°C. One hour later the second measurement show that the core temperature of the body was 25°C. assume that the time of death corresponds to t = 0 and that the core temperature at the time was 37°C. determine how many hours elapsed before the body was found. [ Hint: Let t1 > 0 denote the time that the body was discovered.] (Round your answer to one decimal place.)

_____ hr

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)

Assume that GDP (Y ) is 5,000 in a closed economy. Consumption (C) is given by the equation C = 1,200+0.3(Y −T)−50r, where r is the real interest rate, in percent. Investment (I) is given by the equation I = 1, 500 − 50r. Taxes (T ) are 1,000, and government spending (G) is 1,500.

(a) What are the equilibrium values of C, I, and r?

(b) What are the values of private saving, public saving, and national saving? (

(c) For the given consumption function, what does the relationship between consumption and the interest rate imply about the saving schedule?