MathExam

Questions

18.Vijay wants to start a retirement account that will have $460,000 init when he retires in 15 years. How much should he invest every sixmonths in his account to do this if interest is 15% compoundedsemiannually?

Working

Inthis case, the equation formed is a geometric progression with thefirst term being P (1 + 0.075) and the last term is P (1 + 0.075) ^30.

Thiscan be rewritten as 1.075P (where P stands for the amount being addedevery six months). 0.75 is derived from dividing the rate by two. 15%divided by two is 7.5% (0.075)

1.075P + 1.075^{2}P+ 1.075^{3}P+ 1.075^{4}P+ 1.075^{5}P+ 1.075^{6}P+ 1.075^{7}P+ 1.075^{8}P+ 1.075^{9}P+ 1.075^{10}P+ 1.075^{11}P+ 1.075^{12}P+ 1.075^{13}P+ 1.075^{14}P+ 1.075^{15}P+ 1.075^{16}P+ 1.075^{17}P+ 1.075^{18}P + 1.075^{19}P + 1.075^{20}P + 1.075^{21}P + 1.075^{22}P+1.075^{23}P + 1.075^{24}P + 1.075^{25}P + 1.075^{26}P + 1.075^{27}P + 1.075^{28}P + 1.075^{29}P + 1.075^{30}P

Thisis a geometric progression, with the ratio of 1.075, a = 1.075 and itrepeats itself for 30 terms.

Thisgives a total of 103.70240 P which is equal to $460,000

Dividingboth sides by 103.70240 gives the value of P as $4435.77

=$4435.77 (choice A)

19.According to the IMS Institute for Healthcare Informatics, "Useof Medicines in the United States: Review of 2011", Americansspent $8.5 billion on antipsychotic drugs in 2004 an amount whichclimbed to $18.3 billion in 2011.

a)Which of the following linear equations could be used to predictannual American expenditure (in US $ billions) on antipsychotic drugs“y” in a given year “x”, where x = 0 represents the year2004?

Working

Changein y = 18.3 – 8.5 = 9.8

Changein x = 2011 – 2004 = 7

Gradient= 9.8 / 7 = 1.4

Yintercept is the value of y when x = 2004 which is $8.5

Therefore,the equation is y = 1.4x + 8.5 (choice d)

b)Use the equation from part (a) to predict the amount (in US $billions) that Americans will spend on antipsychotic drugs in theyear 2020. Round answer to nearest tenth of a billion dollars.

Working

y= 1.4x + 8.5

2020– 2004 = 16 years

y= (1.4 * 16) + 8.5

y= 22.4 + 8.5

=$30.9

=$31 (Rounded off to the next billion)

c)Fill in the blanks to interpret the slope of the equation: The rateof change of annual spending on antipsychotic drugs with change intime is $1.4 per year.

24. A local car rental agency charted daily demand as shown in thefollowing table:

Number of customers |
6 |
8 |
10 |
12 |
14 |

Probability |
0.15 |
0.20 |
0.25 |
0.30 |
0.10 |

Findthe expected number of customers.

Working

Probabilityof 6 customers showing is 0.15. So 0.15 * 6 = 0.9 (one customer)

Probabilityof 8 customers is 0.20. Therefore, 0.20 * 8 = 1.6 (two customers)

Probabilityof 10 customers showing is 0.25. So 0.25 * 10 = 2.5 (three customers)

Probabilityof 12 customers showing is 0.3. So 0.3 * 12 = 3.6 (four customers)

Probabilityof 14 customers showing is 0.10. So 0.1 * 14 = 1.4 (two customers)

Thetotal number of expected customers is 1 + 2+ 3 + 4 + 2 = 12 customers

Expectednumber of customers is therefore, 12 customers.