A 24-mH inductor, an 8.1-Ω resistor, and a 5.9-V battery are connected in series as in the figure below. The switch is closed at t = 0. A circuit contains a battery, a switch, an inductor, and a resistor. The circuit starts at the positive terminal of the battery labeled emf ℰ, goes through the switch labeled S, goes through the inductor labeled L, goes through the resistor labeled R and ends at the negative terminal of the battery. 

(a) Find the voltage drop across the resistor at t = 0. V 

(b) Find the voltage drop across the resistor after one time constant has passed. V 

(c) Find the voltage drop across the inductor at t = 0. V 

(d) Find the voltage drop across the inductor after one time constant has elapsed. V

1) Contract Law - Insurance

Mario has purchased a new kart, and would like to get some insurance on it. The kart is

worth $1,000, and there is a 10% chance he will be in an accident and do $500 worth of

damage to his kart. Mario's utility function is U(V) = ln(V), where V is the value of his

kart.

a) Calculate Mario's expected utility with no insurance.

b) Now suppose Mario can buy auto insurance from Yoshi for a premium of $I that will completely compensate him for his $500 loss if it occurs. Find an expression Mario's

expected utility with insurance.

c) What is the most Mario is willing to pay for insurance?

d) Repeat steps a) - c) with a utility function of U(V) = V. Comment on why any differences are present in willingness to pay for insurance.

1. Prepare a summary of 150-250 words after watching video resources related to coordinate system, map projection and datum

Coordinate Map Projection-Video Resources

Note: For your ease and to avoid cumbersome downloading activity due to internet connectivity, I have chosen these video tutorials explaining different concepts. Please watch these as a part of your lecture notes.

Map Projections Explained - A Beginners Guide

https://www.youtube.com/watch?v=wlfLW1j05Dg

Introducing Coordinate Systems and Map Projections

https://www.youtube.com/watch?v=PICwxT0fTHQ

Map Projections

https://www.youtube.com/watch?v=nJ5r4HJMrfo

Everything you would want to know about Map Projections!

https://www.youtube.com/watch?v=dHT0ckOfScU

Different map projections

https://www.youtube.com/watch?v=qM87BDtXSSQ

A force F = −F0 e ^−x/λ (where F0 and λ are positive constants) acts on a particle of mass m that is initially at x = x0 and moving with velocity v0 (> 0). Show that the velocity of the particle is given by

v(x)=(v0^2+(2F0λ /m)((e^-x/λ)-1))^1/2

where the upper (lower) sign corresponds to the motion in the positive (negative) x direction. Consider first the upper sign. For simplicity, define ve=(2F0 λ /m)^1/2 then show that the asymptotic velocity (limiting velocity as x → ∞) is given by v∞=(v0^2-ve^2)^1/2 Note that v∞ exists if v0 ≥ ve.Sketch the graph of v(x) in this case. Analyse the problem when v0 < ve by taking into account of the lower sign in the above solution. Sketch the graph of v(x) in this case. Show that the particle comes to rest (v(x) = 0) at a finite value of x given by xm=−λ ln(1-v0^2/ve^2)

1. Boltzmann statistics are used to find the distribution or distribution of the velocity of Inert gas at any temperature. If D (v) is the velocity distribution of inert gas at T, then the probability that atoms (or Molecules) of inert gas have velocity in the dv range is equal to D (v) dv, where
D (v) dv = 4π (m / 2πkT) ^ 3⁄2 (v ^ 2) e ^ (- mv2⁄2kT) dv

2.1 Draw the graph between D (v) and v when the inert gas has a temperature of 1000 K (Recommended: Use a program such as Mathematica) to explain. Graph style
2.2 In the Thermosphere atmosphere, which is 100 - 150 km above the earth, the temperature is around 1000 K. Find the probability
Is that the nitrogen gas molecules will escape from gravity. In which the molecules must be faster than the velocity From the earth's surface, which is equal to 11 km / s (recommended: for integration D (v) dv, use the program For example Mathematica)
2.3 The lunar surface velocity is 2.4 km / s. Find the probability that the nitrogen gas molecules will escape from the force. Gravity of the moon And explain that Why does the moon have no atmosphere? (Recommended: set the temperature of the moon's surface to equal1000 K)

5. Prove the Following:

a. Let {v1, . . . , vn} be a finite collection of vectors in a vector space V and suppose that it is not a linearly independent set.

i. Show that one can find a vector w ∈ {v1, . . . , vn} such that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that Span(S) = Span(v1, . . . , vn).

ii. Suppose T ⊂ {v1, . . . , vn} is known to be a linearly independent subset. Argue that the vector w from the previous part can be chosen from the set {v1, . . . , vn} \ T.

b. Let V be a vector space and v ∈ V a vector in it. Argue that the set {v} is a linearly independent set if and only if v 6= ~0. Then use this fact together with part i of part a to prove that if {v1, . . . , vn} is any finite subset of V containing at least one non-zero vector, you can obtain a basis of Span(v1, . . . , vn) by simply discarding some of the vectors vi from the set {v1, . . . , vn}.

c. Suppose {v1, . . . , vn} is a linearly independent set in V and that {w1, . . . , wm} is a spanning set in V.

i. Prove that n ≤ m. Hint: use part ii of part a to argue that, for any r ≤ min(m, n), there is a subset T ⊂ {w1, . . . , wm} of size r such that {v1, . . . , vr , w1, . . . , wm} \ T is a spanning set. Then consider the two possibilities when r = min(m, n).

ii. Conclude that if a vector space has a finite spanning set, then any two bases are finite of equal length. (Necessarily, this means that our notion of dimension from class is well-defined and any vector space with a finite spanning set hence has finite dimension).

Comparative balance sheets for 2015 and 2016 and a statement of income for 2016 are given below for Metagrobolize Industries. Additional information from the accounting records of Metagrobolize also is provided. METAGROBOLIZE INDUSTRIES Comparative Balance Sheets December 31, 2015 and 2016 ($ in 000s)

2015 2016

Assets

Cash $ 120 $ 97

Petty cash 6 5

Accounts receivable 28 16

Allowance for uncollectible accounts (5) (3)

Investment in A Corp 15 0

Investment in B Inc 0 22

Inventory 23 33

Prepaid insurance 6 5

Supplies 10 13

Land 75 60

Building 120 145

Less: Accumulated depreciation (24) (30)

Equipment 60 40

Less: Accumulated depreciation (36) (28)

$ 398 $ 375

Liabilities

Accounts payable $ 38 $ 33

Salaries payable 7 9

Interest payable 2 4

Note payable 160 120

Owner's Equity

Owner's contribution   145 176

Capital 46 62

Withdrawal 0 (29)

$ 398 $ 375

METAGROBOLIZE INDUSTRIES Income Statement For the Year Ended December 31, 2016

Revenues

Sales revenue $ 192

Expenses

Cost of goods sold $ 94

Bad debt expense 3

Insurance expense 26

Supplies expense 22

Depreciation expense 11

Salaries expense 24

Interest expense 12

Utilities expense 11

Total expenses 203

Income from operations (11)

Other income and expenses

Gain of sale of land 22

Gain on sale of equipment 7

Loss on sale of Investment A (2)

Total other income and expenses 27

Net income $ 16

Required: Prepare the statement of cash flows of Metagrobolize for the year ended December 31, 2016. Present cash flows from operating activities by the direct method.

Comparative balance sheets for 2016 and 2015 and a statement of income for 2016 are given below for Metagrobolize Industries. Additional information from the accounting records of Metagrobolize also is provided. METAGROBOLIZE INDUSTRIES

Comparative Balance Sheets December 31, 2016 and 2015 ($ in 000s)

                   2016 2015

Assets

Cash           $ 460    $ 190

Accounts receivable 470    250

Inventory    620                 380

Land              625 575

Building          800                800

Less: Accumulated depreciation (200) (170)

Equipment      2,800 2,510

Less: Accumulated depreciation (451) (420)

Patent             1,600          1,800

                       $ 6,724 $ 5,915

Liabilities

Accounts payable $ 720     $ 470

Accrued expenses payable 210 165

Lease liability—land     150      0

Shareholders' Equity

Common stock 2,650    2,500

Paid-in capital—excess of par    500     470

Retained earnings 2,494      2,310

                               $ 6,724 $ 5,915

METAGROBOLIZE INDUSTRIES

Income Statement For the Year Ended December 31, 2016 ($ in 000s)

Revenues

Sales revenue $ 2,807

Gain on sale of land 60

                                   $ 2,867

Expenses

Cost of goods sold $ 920

Depreciation expense—building 30

Depreciation expense—equipment 328

Loss on sale of equipment 10

Amortization of patent 200

Operating expenses $ 600                     2,088

Net income $ 779

Additional information from the accounting records:

a. During 2016, equipment with a cost of $330,000 (90% depreciated) was sold.

b. The statement of shareholders’ equity reveals reductions of $180,000 and $415,000 for stock dividends and cash dividends, respectively. Required: Prepare the statement of cash flows for Metagrobolize Industries using the indirect method.

Comparative balance sheets for 2016 and 2015 and a statement of income for 2016 are given below for Metagrobolize Industries. Additional information from the accounting records of Metagrobolize also is provided. METAGROBOLIZE INDUSTRIES

Comparative Balance Sheets

December 31, 2016 and 2015 ($ in 000s)

2016 2015

Assets Cash $ 470 $ 160

Accounts receivable 490 260

Inventory 640 385

Land 650 590

Building 800 800

Less: Accumulated depreciation (250) (230)

Equipment 2,850 2,570

Less: Accumulated depreciation (472) (440)

Patent 1,700 1,950 $ 6,878 $ 6,045

Liabilities

Accounts payable $ 740 $ 440

Accrued expenses payable 220 155

Lease liability—land 150 0

Shareholders' Equity

Common stock 2,600 2,500

Paid-in capital—excess of par 600 515

Retained earnings 2,568 2,435

$ 6,878 $ 6,045

METAGROBOLIZE INDUSTRIES

Income Statement

For the Year Ended December 31, 2016 ($ in 000s)

Revenues

Sales revenue $ 2,839

Gain on sale of land $ 65

$ 2,904

Expenses

Cost of goods sold $ 940

Depreciation expense—building 20

Depreciation expense—equipment 356

Loss on sale of equipment 10

Amortization of patent 250

Operating expenses $ 600

2,176

Net income $ 728

The comparative balance sheets for 2016 and 2015 and the income statement for 2016 are given below for Arduous Company. Additional information from Arduous’s accounting records is provided also.

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ARDUOUS COMPANY
Income Statement