## Wednesday, 25 April 2012

### Functional Analysis MIT question/problem

Q: Consider a real-valued function f : R −→ R which is locally integrable in the
sense that
(5.2) gL(x) =
f(x) x ∈ [−L,L]
0 x ∈ R \ [−L,L]
is Lebesgue integrable of each L ∈ N.

A:

f: R ---> R
locally integrable

(5.2) Greenland[gL](x) =
function (x) x is an element of [-L,L]
0 x is an element of R divided by [-L,L]

Lebesgue integrable = L is an element of  N

5.2gL f (R) =
L = R + L + (- L)

A little confused on this.

Thanks.

### Set Theory -Calculus Question.

Write carefully using English and using it correctly.
You have 2 hours and 30 minutes.

1. Show that if q Î Q is a square in Q, then 2q is not a square in Q.

q equals q

=
# Q #
=

=   =
#Q# #Q# = 2Q
=   =

The above is not a square in Q rather a set of squares in Q. =).

Thanks.

Think this is incorrect?

### Trigonometry Question.

Q:

Sin B = L of opposite side
--------------------
L of Hypotenuse

Tan B = L of opposite side
----------------------

Sin B = 50/14 - Approx - 3.6
Tan B = 50/48 - Approx - 1.02

Sin = 50/14
50 = 14 * sin = 3.6
Sine = 3.6
+ on going below....

Sin B = 14,28,42,49 {3.5},50 3.6

Sin B = 24/25 - Approx - 0.95
Tan B = 24/7 - Approx - 3.4

Sin B = 7/25 - Approx - 3.6
Tan B = 7/24 - Approx - 3.4

Sin B = 7..24 - Approx - 3.4
Tan B = 7/245 - Approx - 3.6

Think this is incorrect?

Thanks

## Monday, 23 April 2012

### {Decision Making} - CVP Analysis - Capacity Usage of a Leisure Centre

A local government authority owns and operates a leisure centre with numerous sporting facilities, residential accommodation, a cafeteria and a sports shop. The summer season lasts 20 weeks including a peak period of six weeks corresponding to the school holidays. The following budgets have been prepared for the next summer season:

Accommodation
60 single rooms let on a daily basis
35 double rooms let on a daily basis at 160 per cent of the single room rate.
Fixed costs £29,900
Variable costs £4 per single room per day and £6.40 per double room per day.
Sports Centre
Residential guests each pay £2 per day and casual visitors £3 per day for the use of facilities.
Fixed costs £15,500
Sports Shop
Estimated contribution £1 per person per day.
Fixed costs £8250
Cafeteria
Estimated contribution £1.50 per day.
Fixed costs £12,750

During the summer season the centre is open seven days a week and the following activity levels are anticipated.

Double rooms fully booked for the whole season.

Single rooms fully booked for the peak period but at only \$0 per cent of capacity during the rest of the season.
30 casual visitors per day on average.

Questions -
a): Calculate the charges for the single and double bedrooms assuming that the authority wishes to take £10,000 profit on accommodation.
b): Calculate the anticipated total profit for the leisure centre as a whole for the season.

Ans -
a): £4 * 60 = £240
£6.40 * 35 = £224 (approx)

£240 * 30 * 7 = £7200 {*7} = £50400  {1 week}
£224 * 35 * 7 = £7800 {*7} = £62400 {1 week}

Total for 1 week = £112,800
Minus £10,000 profit = £102,800

Equals = £102,800

£112,800 * 20 = £2,256,000 {Total Profit}
Fixed Costs = £29,900
£2,256,600
- £   29,900
=  2,226,700
-     10,000
=  2,216,700
Total Charges : 39,900
Total Profit: 2,216,700

b): RVisitors: £2 * 7 = £14
CVisitors: £3 * 7 = £21

£14 * 30 = £420
£21 * 30 = £630

£420 * 30 = 12600
£630 * 30 = 19200

Total: 12,600 + 19,200 = 31,800
£10,000 - Accommodation Authority Fee
£15,500 - Fixed Costs
= £6,300 {Total Profit Remaining}

Thank you very much =)

### Clinical Professional Decision Making Question

3. Construct a five-year leadership plan for yourself, then in one-page to two-page paper, outline a development plan based on your ultimate health leadership position goal. How long will it take you to read your ultimate goal?

Year 1: Reduce Meds - 5ml - 23
Year 2: Reduce Meds - 5ml - 24
Year 3: Reduce Meds II - 50mg Jan - 25
Year 3: Reduce Meds II - 50mg Dec - 25
Year 4: Reduce Meds II - 50 mg Jan - 26
Year 5: Meds II - 50mg March - 27
Year 5: Meds II - Steady - 27+

This was an example, particularly difficult, but enabled to profess the correct analysis required for Clinical Management of the medication.

Think this is incorrect or have a solution?
Let me know....

Thanks,

## Sunday, 15 April 2012

### Probability Distributions - A Level Math

1. Two fair dice are thrown. If the scores are unequal, the larger of the two scores is recorded. If the scores are equal then that score is recorded. Let X denote the number recorded.
a) show that P(X = 2) = 1/12 and draw up a table showing the probability distribution of X

Ans:
Two fair dice {6 sided}

Scores - unequal:

1. 1+1=1 {X =1}
2. 2+1=3 {X = 2}
3. 1+2=3 {X = 2}
4. 2+2=4 {X = 2]
5. 2+2=4 {X = 2}
6. 2+3=5 {X = 3}
7. 3+2=5 {X = 3}
8. 3+3=6 {X = 3}
9. 3+3=6 {X = 3}
10. 3+4=7  {X = 4}
11. {X = 4}
12. {X = 4}
13. {X = 4}
14. {X = 5}
15. {X = 5}
16. {X = 5}
17. {X = 5}
18. {X = 6}
19. {X = 6}
20. {X = 6}
21. {X = 6}
etcetera...

P(X=2)

X=2 == 4/21 if 1/12 then {1.9/21} e.g.
Calculation Based On: 2/21 (Equal)
0.95/21 = 0.045[recurring]

Calculation Based On: 2/21 (Unequal)

0.95/21 = 0.045[recurring]
Thank you.

Think this is incorrect??

Thanks.

### Probability - Dice

A pair of dice is rolled. What is the probability of getting a sum of 2?

None of the above.

Ans:

{2 Dice}
1. 1+1 = 2
2. 1+1=2
3. 1+2=3
4. 2+1=3
5. 2+2=4
6. 2+2=4
7. 2+3=5
8. 3+2=5
9. 3+3=6
10. 3+3=6
11. 3+4=7
12. 4+3=7
13. 4+4=8
14. 4+4=8
15. 4+5=9
16. 5+4=9
17. 5+5=10
18. 5+5=10
19. 6+5=11
20. 5+6=11
21. 6+6=12
22. 6+6=12

2/22 = 1/11

Ans: None of the above.

Think this is incorrect???