F. 4 maths

2015-06-17 5:37 pm

David is 8 years old when his parents start an education fund. They deposit$450 at the end of every 3 months in a fund that pays 8%/a compoundedquarterly.a.How old is David when the fund is worth $20000?b.How much less time would it take to build the fund to $20000 if theregular deposit were $550?

更新1:

the answer is 16 years old.

回答 (2)

polarbearhmh

2015-06-17 8:19 pm

✔ 最佳答案

(a)

In 3n months, the fund
= 450×(1+8%/4)ⁿ + 450×(1+8%/4)ⁿ⁻¹ + 450×(1+8%/4)ⁿ⁻² +...+ 450×(1+8%/4) + 450
= 450×(1.02)ⁿ + 450×(1.02)ⁿ⁻¹ + 450×(1.02)ⁿ⁻² +...+ 450×(1.02) + 450
= 450(1-1.02ⁿ)/(1-1.02)

Therefore set 450(1-1.02ⁿ)/(1-1.02) = 20000
450×(1.02ⁿ-1)/0.02 = 20000
1.02ⁿ-1 = 8/9
1.02ⁿ = 17/9
n = (log 17 - log 9)/log(1.02)
n = 32.1 (cor. to 3 sig. fig.)
∴ When n = 33, the fund is worth $20000.
∴ When David is 19 years old, the fund is worth $20000.

(b)

Cosider 550(1-1.02ᵐ)/(1-1.02) = 20000
1.02ᵐ = 19/11
m = (log 19 - log 11)/log(1.02)
m = 27.6 (cor. to 3 sig. fig.)
∴ It would take 5 months less to build the fund to $20000.

2015-06-17 19:56:00 補充：
更正

(b)
∴ It would take 5×3 = 15 months less to build the fund to $20000.

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少年時

2015-06-18 2:03 am

Five quarter months.

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