Differential Geometry

Open & Closed Sets, Continuity

Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are continuous

Bonita Ranch & Farm is a distributor of ranch and farm equipment. Its products include small tools, power equipment for trench-digging and fencing, grain dryers, and barn winches. Most products are sold direct via its company Internet site. However, given some of its specialty products, select farm implement stores carry Bonita’s products. Pricing and cost information on three of Bonita’s most popular products are as follows.

Respond to the requirements related to the following independent revenue arrangements for Bonita Ranch & Farm. IFRS is a constraint.

1. On January 1, 2020, Bonita sells augers to Mills Farm & Fleet for $52,800. Mills signs a six-month note at an annual interest rate of 12%. Bonita allows Mills to return any auger that it cannot use within 60 days and receive a full refund. Based on prior experience, Bonita estimates that 5% of units sold to customers like Mills will be returned (using the most likely outcome approach). Bonita’s costs to recover the products will be immaterial, and the returned augers are expected to be resold at a profit. Prepare the journal entries for Bonita on January 1, 2020.

2.On August 10, 2020, Bonita sells 17 mini-trenchers to a farm co-op in western Canada. Bonita provides a 4% volume discount on the mini-trenchers if the co-op has a 15% increase in purchases from Bonita compared with the prior year. Given the slowdown in the farm economy, sales to the co-op have been flat, and it is highly uncertain that the benchmark will be met.

3. Bonita sells three grain/hay dryers to a local farmer at a total contract price of $50,000. In addition to the dryers, Bonita provides installation, which has a stand-alone sales value of $1,020 per unit installed. The contract payment also includes a $1,530 maintenance plan for the dryers for three years after installation. Bonita signs the contract on June 20, 2020, and receives a 20% down payment from the farmer. The dryers are delivered and installed on October 1, 2020, and full payment is made to Bonita.

Prepare the journal entries for Bonita in 2020 related to this arrangement as well as any adjusting journal entries at its December year end

4. On April 25, 2020, Bonita ships 110 augers to Farm Depot, a farm supply dealer in Alberta, on consignment. By June 30, 2020, Farm Depot has sold 70 of the consigned augers at the listed price of $1,320 per unit. Farm Depot notifies Bonita of the sales, retains a 8% commission, and remits the cash due to Bonita.

Prepare the journal entries for Bonita and Farm Depot for the consignment arrangement

Stellar Ltd prepares accounts to 31March every year. Its latest trial balance for the year ended 31 March 2020 is provided below.

Stellar Ltd Trial Balance as at 31 March 2020

Additional information as at 31March 2020 is provided below:

1. Inventories at close of business on 31 March 2020 was valued at £17,500,000 at cost.
2. A cash dividend of £0.10 per share was paid to ordinary shareholders on 27 March 2020. No entries have been made in the accounts for this transaction.
3. Due to the contractual obligation to pay preference dividends, the company recognises and accounts for preference shares as a liability.
4. Depreciation is to be provided for the year ending 31March 2020 as follows:
   1. Property at 1% per annum on cost.
   2. Plant and equipment at 5% per annum on a reducing balance basis.
   3. The depreciation charge for the year is to be apportioned to administrative and distribution expenses as per the table below:

5. Interest on the debentures has not yet been paid and needs to be accrued for the year.
6. To be prudent, the directors wish to create an allowance for receivables equal to 1% of trade receivables. It is company policy to classify all bad debts and any allowances for receivables as distribution expenses.
7. Staff costs outstanding at the financial year end amounted to £500,000 and other expenses included £300,000 which had been paid in advance. Both these expenses are chargeable 60% to administration and 40% to distribution.
8. The amount for rent in the trial balance above relates to the period 1 April 2019 to May 2020. Rent expense is charged 30% to administration and 70% to distribution.
9. Half of the general expenses relate to administration and half to distribution.
10. The corporation tax charge is to be provided at 20% of profits after charging all expenses and interest

Prepare the Statement of Profit and Loss, the Statement of Changes in Equity and the Statement of Financial Position of Stellar Ltd for the financial year end 31 March 2020. (You should show all your workings).

Stellar Ltd prepares accounts to 31March every year. Its latest trial balance for the year ended 31 March 2020 is provided below.

Stellar Ltd Trial Balance as at 31 March 2020

Additional information as at 31March 2020 is provided below:

1. Inventories at close of business on 31 March 2020 was valued at £17,500,000 at cost.
2. A cash dividend of £0.10 per share was paid to ordinary shareholders on 27 March 2020. No entries have been made in the accounts for this transaction.
3. Due to the contractual obligation to pay preference dividends, the company recognises and accounts for preference shares as a liability.
4. Depreciation is to be provided for the year ending 31March 2020 as follows:
   1. Property at 1% per annum on cost.
   2. Plant and equipment at 5% per annum on a reducing balance basis.
   3. The depreciation charge for the year is to be apportioned to administrative and distribution expenses as per the table below:

5. Interest on the debentures has not yet been paid and needs to be accrued for the year.
6. To be prudent, the directors wish to create an allowance for receivables equal to 1% of trade receivables. It is company policy to classify all bad debts and any allowances for receivables as distribution expenses.
7. Staff costs outstanding at the financial year end amounted to £500,000 and other expenses included £300,000 which had been paid in advance. Both these expenses are chargeable 60% to administration and 40% to distribution.
8. The amount for rent in the trial balance above relates to the period 1 April 2019 to May 2020. Rent expense is charged 30% to administration and 70% to distribution.
9. Half of the general expenses relate to administration and half to distribution.
10. The corporation tax charge is to be provided at 20% of profits after charging all expenses and interest

Prepare the Statement of Profit and Loss, the Statement of Changes in Equity and the Statement of Financial Position of Stellar Ltd for the financial year end 31 March 2020. (You should show all your workings).

Waterway Company is a manufacturer of smart phones. Its controller resigned in October 2020. An inexperienced assistant accountant has prepared the following income statement for the month of October 2020.

Prior to October 2020, the company had been profitable every month. The company’s president is concerned about the accuracy of the income statement. As her friend, you have been asked to review the income statement and make necessary corrections. After examining other manufacturing cost data, you have acquired additional information as follows.

1. Inventory balances at the beginning and end of October were:

2. Only 75% of the utilities expense and 60% of the insurance expense apply to factory operations. The remaining amounts should be charged to selling and administrative activities.

Prepare a schedule of cost of goods manufactured for October 2020.

Prepare a correct income statement for October 2020.

Entity A is a listed company that operates the cruise ship business. One of the cruise ships was purchased on 1 Oct 2011. This cruise ship is made up of three main components: (1) cruise’s fabric, (2) cabins and entertainment area and (3) fittings propulsion system.

Details of the cost of its components and their estimated useful lives are as below:

Components Original cost Depreciation basis

(1) Cruise’s fabric (hull, decks, etc.) HK$37,500,000 50 years straight-line

(2) Cabins and entertainment area fittings HK$18,750,000 15 years straight-line

(3) Propulsion system HK$12,500,000 useful life of 80,000 hours

On 30 Sep 2019, no further capital expenditure had been incurred on the cruise ship.

In the year ended 30 Sep 2019, the cruise had experienced a high level of engine trouble, which had cost Entity A considerable revenue loss and compensation costs.  The measured expired life of the propulsion system on 30 Sep 2019 was 50,000 hours. Due to the unreliability of the engines, a decision was made by Entity A on 1 Oct 2019 to replace the whole of the propulsion system at a cost of HK$17,500,000. The old propulsion system was also sold to a second-hand machinery shop with a loss on disposal of $4,250,000. The cash from the disposal was received on 20 Oct 2019.  The expected life of the new propulsion system was 160,000 hours and in the year ended 30 Sep 2020, the cruise had used its engines for 10,000 hours.

At the same time as the propulsion system replacement, Entity A took this opportunity to upgrade the cabin and entertainment facilities at a cost of HK$7,500,000 and repaint the cruise’s fabric at a cost of HK$2,500,000 respectively. After the upgrade of the cabin and entertainment area fittings, it was estimated that their remaining useful life was 10 years.

For calculating depreciation, all the works on the cruise can be assumed to have been completed on 1 Oct 2019. All residual values can be taken as NIL.

REQUIRED:

(1) Measure the depreciation expense of the Cruise’s Fabric for the year ended 30 Sep 2020.

Answer = $

(2) Measure the depreciation expense of the Cabins and entertainment area fittings for the year ended 30 Sep 2020.

Answer = $

(3) Measure the depreciation expense of the Propulsion system for the year ended 30 Sep 2020.

Answer = $

(4) Measure the carrying amount of the Cruise’s Fabric on 30 Sep 2020.

Answer = $

(5) Measure the carrying amount of the Cabins and entertainment area fittings on 30 Sep 2020.

Answer = $

(6) Measure the carrying amount of the Propulsion system on 30 Sep 2020.

Answer = $

(7) Measure the carrying amount of Entity A’s cruise ship on 30 Sep 2020.

Answer = $

(8) Measure the cash received from the sale of the old propulsion system.

Answer = $

Testing Hypotheses. For the following passages, indicate whether the evidence mentioned is falsifying evidence for the hypothesis or confirming evidence for the hypothesis:::

1. Hypothesis: A duck stole my wallet.

Evidence: I find my wallet in the seat of my locked car.

2. Hypothesis: A duck stole my wallet.

Evidence: I find my wallet by the fountain where ducks gather, with indentions shaped like the bill of a duck.

3. Hypothesis: My friend Paul studied for the exam last weekend.

Evidence: Sarah, the librarian, says she saw Paul enter the library at noon on Saturday, and not leave until 3.

4. Hypothesis: My friend studied for the exam last weekend.

Evidence: Jessica says she saw Paul at a Party Saturday night, and recall that he did not go home until late Sunday afternoon, after the library had closed.

In the late 1970s, there was a major controversy over a dam built by the Tennessee Valley Authority called the Tellico Dam, which is located near the mouth of the Little Tennessee River. After construction was completed and almost all of the land purchases (or evictions) had been done, the Supreme Court ruled that it could not go on because of the presence in the Little Tennessee of a fish on the endangered species list called the Snail Darter. (Congress later voted to exempt the dam from the law and the dam was closed in the late fall of 1979, creating Tellico Lake.) While the economic benefits of the dam were admittedly dubious, one of the common arguments for completing the project was this: "We have spent more than $100 million to build this dam. We need to complete it because we will have wasted all this money otherwise." Please evaluate this argument from an economic point of view.

In a factory in which metal parts are degreased with organic solvents, the intern breaks a bottle containing a mixture of 5 liters of tetrachloroethylene (PER, MW = 166 g/mol, ρ = 1.62 g/cm3 , Pv = 18 mm Hg) and 10 liters of trichloroethene (TCE, MW = 131 g/mol, ρ = 1.46 g/cm3 , Pv = 54 mm Hg) in a closed room at 20oC. The room has a total volume of 50 m3 . Assume PER and TCE form an ideal mixture in the liquid phase.

a. What is the composition (in mole fractions) of the liquid mixture in the bottle?

b. What are the equilibrium concentrations of PER and TCE in the air of the room in mg/L? Note: Make sure to use Raoult’s Law. [PER] = 49.9 mg/L

c. What is the composition (in mole fractions) of the remaining liquid on the floor after equilibrium? XPER = 0.790