MATH107Quiz5SOLNS.pdf - MATH 107 QUIZ 5 SOLUTIONS October,...

MATH 107 QUIZ 5 SOLUTIONS
October, 2018 Instructor: S. Sands
1. (4 pts) Which of these graphs represent a one-to-one function? Answer(s): A
( no explanation required .) (There may be more than one graph that qualifies.)
(A)
No vertical or horizontal line
crosses the graph more than
once, so the graph represents
a function, and also is a one-
to-one function.
(B)
Not a function (vertical
line crosses graph more
than once), so not a one-
to-one function.
(C)
The horizontal line y = − 3
intersects the graph more than
once
(D)
There are horizontal
lines crossing the graph
more than once.
2. (6 pts) Based on data about the growth of a particular species of trees, the following logarithmic model
about these trees was determined:
h ( t ) = 6.2(ln( t ) + 1),
where t = age of tree in years and h ( t ) = height of tree, in feet.
(Note that "ln" refers to the natural log function) (explanation optional)
Using the model,
(a) At age 5 years, how tall is this type of tree, to the nearest tenth of a foot?
h (5) = 6.2 ( ln(5) + 1)  6.2 (2.6094)  16.2
The model estimates that the 5-year-old tree will be 16.2 feet tall.
(b) At age 20 years, how tall is this type of tree, to the nearest tenth of a foot?
h (20) = 6.2 ( ln(20) + 1)  6.2 (3.9957)  24.8
The model estimates that the 20-year-old tree will be 24.8 feet tall.
3. (4 pts) Convert to a logarithmic equation: 8
x
= 128.
(no explanation required)
3. C
A. log
?
128 = 8
B. log
8
? = 128
C. 𝐥𝐨𝐠
𝟖
??𝟖 = 𝒙 The base is 8. The value of the log is the exponent x.
D. log
?
8 = 128

4.
(8 pts)
Solve the equation. Check all proposed solutions.
Show work in solving and in checking, and state your
final conclusion. [section 0.9, 5.3]
√44 − ? = 2 − ?
(√44 − ? )
2
= (2 − ?)
2
Square both sides.
44 – x = 4 – 4 x + x 2
0 = x 2 – 3 x – 40
0 = ( x – 8)( x + 5)
x – 8 = 0 or x + 5 = 0
x = 8 or x = – 5. Potential solutions
Check:
x = 8: x = – 5:
√44 − 8
? 2 − 8
√44 − (−5)
? 2 − (−5)
√36 ? −6 √49 ? 7
6 ≠ −6 7 = 7
The equation is NOT satisfied.
x = 8 is NOT a solution. x = – 5 IS a solution.
x = – 5 is the only solution.
5. (8 pts)
(a)
log
3
1 = 0
because 3 0 = 1. For any base b ,
log 1
b
= 0.
(b) Let
? = log
3
1
√3
State the exponential form of the equation.
?
𝒙
=
1
√3
(c) Determine the numerical value of
log
3
1
√3
in simplest form. Work optional.
Since
1
√3
=
1
3
1/2
= 3
−1/2
, we have
3
?
= 3
−1/2
We raise 3 to the – 1/2 power in order to get
1
√3
.
x = – 1/2, so
? = log
3
1
√3
= −
?
?
Alternatively,
log
3
1
√3
= log
3
1 − log
3
√3
= 0 − log
3
3
1
2
= −
1
2
log
3
3 = −
1
2
(1) = −
?