數學問題 40points

hihi

2009-10-31 6:06 am

1.The value of a new car is $300000 and its value is $180000 after two years. It is known that the annual depreciation rate decreases gradually. If the difference of the depreciation rates of the first two years is 5%,find the depreciation rate of the first year.

2.Mr Wong deposits $20000 in a bank for two years and the interest is calculated once a year. If the annual compond interest rate is the same as the annual simple interest rate,and that the amount calculated by compound interest is $30 more than the amount calculated by simple interest,find the annual interest rate,correct to 2 decimal places.

3.A bank offers a large sum fixed plan of one-year period. the interest is calculated and compounded half-yearly. The annual interest rate for each 6-month period increase gradually. Moreover, the difference of the annual interest rates of two consecutive 6-month periods is 1%. Miss Wong joins the plan by investing $1000000. She receives an amount of $1035300 after one year. Find the annual interest rate for the first half year.

Thank you very much!

回答 (1)

用戶9112

2009-10-31 6:03 pm

✔ 最佳答案

(1) Let the depreciation for the first year be x%
Then the depreciation for the second year is (x - 5)%
300000 * (1 - x%) * (1 - x% + 5%) = 180000
(1 - x / 100)(1 - x/100 + 5/100) = 18/30
(100 - x)(105 - x) = 6000
x^2 - 205x + 10500 - 6000 = 0
x^2 - 205x + 4500 = 0
(x - 180)(x - 25) = 0
x = 180 (rejected) or x = 25
The depreciation for the first year is 25%
(2) Let the annual interest rate be r% , i.e. r/100
The compound interest = 20000 * (1 + r/100)^2 - 20000
The simple interest = 20000 * 2r/100
The difference = 20000 * (1 + r/100)^2 - 20000 - 2r/100) = 30
20000[1 + 2r/100 + r^2/10000 - 1 - 2r/100] = 30
20000[r^2/10000] = 30
2r^2 = 30
r = √15 = 3.87
The interest rate is 3.87%
(3) Let the annual interest rate for the first half year be r%, i.e. r/100
Then the annual iinterest rate for the second half year is (r + 1)/100
Total amount = 1000000 * (1 + r/100/2) * [1 + (r + 1)/100/2] = 1035300
(200 + r)(201 + r) = 41412
r^2 + 401r - 1212 = 0
(r - 3)(r + 404) = 0
r = 3 or r = -404 (rejected)
The annual interest rate for the first half year is 3%

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