F.4 MATHS

1.The value of a machine is $450 000 and its value depreciates by 20% every
year.At least how many years later will its value be less than$100 000?

Let V be the value of the car and t be the time
V=$450000x(1-20%)^t
From V=$100000
$100000=$450000x(1-20%)^t
$100000=$450000x0.8^t
2/9=0.8^t
log2/9=log0.8^t
log2/9=tlog0.8
t=6.740 (corr to 3dp)
So at least 7 years later its value will be less than $100 000.


2. Mr. chan borrows $10 000 from a bank at an interest rate of 12% p.a.
compounded monthly.For at most how many months can Mr. chan borrow
the money so that the total repayment to the bank will not exceed $11 000?

Let V be the amount of money and t be the time
V=$10000x(1+12/12%)^t
From V=$11000
$11000=$10000x(1+12/12%)^t
$11000=$10000x1.01^t
1.1=1.01^t
log1.1=log1.01^t
log1.1=tlog1.01
t=9.579 (corr to 3dp)
So Mr. chan can borrowthe money for at most 9 months

1)Let the value of machine at least y years will be less than $100 000

450 000x(1-20%)^y<100 000
80%^y<100 000÷450 000
0.8^y<2/9
ylog0.8>log(2/9) (＊log0.8係負數)
y>log(2/9)÷log0.8
y>6.74
∴At least 7years later will its value be less than$100 000.



2)Let Mr. chan can borrow the money for y months so that the total repayment to the bank will not exceed $11 000

10 000X{1+(12%÷12)}^y<11 000
(1+1%)^y<11 000÷10 000
1.01^y<1.1
ylog1.01<log1.1
y<log1.1÷log1.01
y<9.58
∴Mr. chan can borrow the money for 9months so that the total repayment to the bank will not exceed $11 000.
(＊因為呢題問"not exceed"所以應該用"大過等於"
但我打唔到"大過等於"個付號所以用左"<")