If a beam of light has an intensity Io is projected vertically downward into
water, then its density I(x) at a depth x is I(x) = Io e-1.4x. At what depth is
the intensity one half its value at the surface?
Solution:
I(x) = Io e-1.4x
when x = 0
I(0) = Io e-0
I(0) = Io
when I(x) = ½ Io
½ Io = Io e-1.4x
½ = e-1.4x
ln ½ = - 1.4x ln e
ln 21
x=
−1.4
x = 0.495
, Refresher - MATHEMATICS Quiz 14
PROBLEM 2:
A machine is to be purchased for P155,000 it has an estimated life of 8 years and
a salvage value of P6000. A sinking fund is to be established so money will be
available to purchase a replacement when the first machine wears out at the end
of 8 years. An amount of P13,030 is to be deposited at the end of each year
during the lifetime of the first machine into this sinking fund. What interest rate (%)
that this fund must earn to produce sufficient fund to purchase the replacement
machine at the end of 8 years?
Solution:
Required amount after 8 years:
F = 155000 – 6000
F = P149000
A ⎡⎣(1 + i)n - 1⎤⎦ = P13030
F=
i F=149000
13030 ⎡⎣(1 + i)8 - 1⎤⎦
149000 =
i
(1 + i)8 - 1
11.435 =
i
Solving for i = 10% (trial and error)
Check:
(1 + 0.10)8 - 1
= 11.435 ok
0.10
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