maths mock question

[匿名]

2013-02-08 2:53 am

1.the profit($y) of investing in a new stock in the t th month is given by y=-16t^2+800t=-16(t-25)^+10000. find the max total profit can be made by investing in this new stock.(given that 1^2+2^2+...+n^2=n(n+1)(2n+1)/6

2.scientists found that there are 4000' white dolphins' on earth this year. according to their research, there are 200 new-born white dolphins per year. they also found that 8% of white dolphins die every year(including the new-born white dolphins in that year)
show that the no. of white dolphins on earth n years later will be
1700(0.92)^n+2300

回答 (1)

用戶2053

2013-02-08 6:54 am

✔ 最佳答案

1)For the profit $y = - 16(t - 25)² + 10000 > 0 ,
(t - 25)² < 625
- 25 < t - 25 < 25
0 < t < 50The max total profit can be made by investing in this new stock for 49 months= Σ(t = 1→49) - 16(t - 25)² + 10000
= - 16 [ (1 - 25)² + (2 - 25)² + ... + (49 - 25)² ] + 49 * 10000
= - 16 [ (-24)² + (-23)² + ... + (-1)² + 0 + 1² + 2² + ... + 24² ] + 490000
= - 32 ( 1² + 2² + ... + 24² ) + 490000
= - 32 ( 24(24+1)(24*2+1)/6 ) + 490000
= $333,200
2)The no. of white dolphins on earth 1 year later
= (4000 + 200)(1 - 8%)2 years later
= ( (4000 + 200)(1 - 8%) + 200 )(1 - 8%)
= (4000 + 200)(1 - 8%)² + 200(1 - 8%)3 years later
= ( (4000 + 200)(1 - 8%)² + 200(1 - 8%) + 200 ) (1 - 8%)
= (4000 + 200)(1 - 8%)³ + 200(1 - 8%)² + 200(1 - 8%) n years later
= (4000 + 200)(1 - 8%)ⁿ + 200(1 - 8%)ⁿ⁻¹ + ... + 200(1 - 8%)
= 4000(0.92)ⁿ + 200(0.92ⁿ + 0.92ⁿ⁻¹ + ... + 0.92)
= 4000(0.92)ⁿ + 200(0.92(1 - 0.92ⁿ)/(1 - 0.92))
= 4000(0.92)ⁿ + 2300(1 - 0.92ⁿ)
= 1700(0.92)ⁿ + 2300

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