Download SOML MEET 4

SOML MEET 4
EVENT 2
Log and Exponential Functions
1.
[2 Points]
Suppose
NAME: _____________
TEAM: _____________
SCHOOL: _____________
log b Y = log c Z = x
Find log 2
and
b
= 2.
c
Y
.
Z
ANS: _____________
2.
[3 Points]
I
,where I is the intensity
S
of the earthquake (measured by the amplitude of a seismography reading
taken 100km from the epicenter of the earthquake) and S is the intensity of
a “standard” earthquake (whose amplitude is 1 micron = 10 −4 cm ).
The 1906 earthquake in San Francisco had an estimated magnitude of 8.3
on the Richter scale. In the same year a powerful earthquake occurred on
the Columbian-Ecuador border and was four times as intense. What was
the magnitude of the Columbia-Ecuador earthquake on the Richter scale,
to the nearest tenth?
The magnitude M of an earthquake is: M = log
ANS: _____________
3.
[5 Points]
An infection strain of bacteria grows by doubling every hour. When a
certain critical number of bacteria are present in the bloodstream, a person
becomes ill. If a single bacterium infects a person, the critical level is
reached in 24 hours. How long will it take for the critical level to be
reached if the same person is infected with 10 bacteria? Give your answer
to the nearest hundredth of an hour.
ANS: ___________hrs
SOML MEET 4
EVENT 2
Log and Exponential Functions
1.
[2 Points]
Solution:
Suppose
NAME: ___KEY______
TEAM: _____________
SCHOOL: _____________
log b Y = log c Z = x
and
Y
b
= 2 . Find log 2 .
c
Z
log 2
Y
= log 2 2 x = x.
Z
log b Y = x ⇒ b x = Y
log c Z = x ⇒ c x = Z
x
Y bx ⎛ b ⎞
= x = ⎜ ⎟ = 2x ;
Z c
⎝c⎠
ANS: _____x______
2.
[3 Points]
The magnitude M of an earthquake is: M = log
I
, where I is the intensity of the
S
earthquake (measured by the amplitude of a seismography reading taken 100km from the
epicenter of the earthquake) and S is the intensity of a “standard” earthquake (whose
−4
amplitude is 1 micron = 10 cm ). The 1906 earthquake in San Francisco had an
estimated magnitude of 8.3 on the Richter scale. In the same year a powerful earthquake
occurred on the Columbian-Ecuador border and was four times as intense. What was the
magnitude of the Columbia-Ecuador earthquake on the Richter scale, to the nearest tenth?
Solution:
If I is the intensity of the San Francisco earthquake, then from the definition of magnitude
we have:
I
M = log
S
= 8.3
The intensity of the Columbian-Ecuador earthquake was 4I, so its magnitude was:
M = log
4.
[5 Points]
4I
= log 4 + 8.3 = 8.9
S
ANS: _____8.9____
An infection strain of bacteria grows by doubling every hour. When a
certain critical number of bacteria are present in the bloodstream, a person
becomes ill. If a single bacterium infects a person, the critical level is
reached in 24 hours. How long will it take for the critical level to be
reached if the same person is infected with 10 bacteria? Give your answer
to the nearest hundredth of an hour.
Solution: B ( t ) = B0 2t . When B0 = 1 , B ( 24 ) = 224 = 16, 777, 216 . When B0 = 10 , we solve
10 ( 2t ) = 16, 777, 216 . 2t = 1, 677, 721.6 → t ln 2 ≈ ln (1, 677, 721.6 ) → t ≈ 20.68 hrs.
ANS: _20.68
hours_