Deck 5: Trigonometric Functions: Right Triangle Approach

Question 1

Sketch the graph of the function. State the domain, range, and asymptote.

Answer 1

Domain:
, Range:
, Asymptote:

Question 2

Sketch the graph of the function. $f ( x ) = 3 + 3 ^ { x }$

Answer 2

Question 3

Find the solution of the equation $3 ^ { x + 2 } = 5 ^ { 2 x }$
, correct to four decimal places.

Answer 3

$x = \frac { 2 \log 3 } { 2 \log 5 - \log 3 } \approx 1.0363$

Question 4

Combine into a single logarithm.

Question 5

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by $t = 3 \frac { \log ( N / 50 ) } { \log 2 }$ Find the time required for the colony to grow to five million bacteria.

Question 6

How long will it take an investment of $\$ 300$ to triple, if the interest rate is $9.5 \%$ per year compounded continuously?

Question 7

Expand the logarithmic expression.

Question 8

Write the equation in logarithmic form. $64 ^ { - 1 / 2 } = \frac { 1 } { 8 }$

Question 9

Expand the logarithmic expression.

Question 10

If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following compounding methods
a)Continuously
b) Daily

Question 11

Evaluate the expression without using a calculator. $\log 50 + \log 2$

Question 12

Solve the equation. Find the exact solution. $3 ^ { 1 - x } = 7$

Question 13

Use a graphing device to find the solutions of the equation, correct to two decimal places. $3 \ln x = 6 - 2 x$

Question 14

Solve the equation. Find the exact solution. $\log x + \log ( x + 1 ) = \log 20$

Question 15

Graph the function, not by plotting points, but by starting from the graph of $y = e ^ { x }$ . State the domain, range, and asymptote. $h ( x ) = e ^ { x + 1 } - 5$

Question 16

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 17

Evaluate the expression without using a calculator. $\log _ { 4 } \left( \frac { 1 } { 64 } \right)$

Question 18

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $\log _ { 2 } 6$

Question 19

Combine into a single logarithm.

Question 20

A culture contains $680$ bacteria. After just $30$ minutes the count is $3250$ . Find the number of bacteria after $3.5$ hours. After how many minutes will the number of bacteria triple?

Question 21

Expand the logarithmic expression.

Question 22

Solve the equation. Find the exact solution. $\log x + \log ( x + 1 ) = \log 12$

Question 23

How much more intense is an earthquake with a magnitude of $5.4$ on the Richter scale than one with a magnitude of $3.2$ ?

Question 24

A substance has a hydrogen ion concentration of $\left[ \mathrm { H } ^ { + } \right] = 3.7 \times 10 ^ { - 6 }$ M. Find the pH of the substance.

Question 25

Expand the logarithmic expression.

Question 26

The velocity of a sky diver t seconds after jumping is given by $v ( t ) = 80 \left( 1 - e ^ { - 02 t } \right)$ . After how many seconds is the velocity 60 ft/s?

Question 27

The intensity level of sound at a crowded football game was measured at $96$
dB. Find the intensity in $\text {watts/m }^2$

Question 28

Combine into a single logarithm.

Question 29

Write the equation in logarithmic form. $81 ^ { - 1 / 2 } = \frac { 1 } { 9 }$

Question 30

Find the solution of the equation $5^{-x/3}=0.63$ , correct to four decimal places.

Question 31

Combine into a single logarithm.

Question 32

Find the domain of the function. $f ( x ) = \ln \left( x ^ { 2 } + 5 x + 6 \right)$

Question 33

Radium - $221$ has a half-life of $30 \mathrm {~s}$ . Suppose we have a $300 \mathrm {~g}$ sample.
(a) Find a formula for the mass remaining after $t$ seconds
b) How much of the sample remains after $200$ seconds?
c) After how long will only $20 \mathrm {~g}$ remain?

Question 34

Evaluate the expression without using a calculator. $\log _ { 7 } ( \sqrt { 7 } )$

Question 35

Evaluate the expression without using a calculator. $\log _ { 5 } 500 - \log _ { 5 } 4$

Question 36

Evaluate the expression without using a calculator. $e ^ { 2 \ln 8 }$

Question 37

Graph the function, not by plotting points, but by starting from the graph of $y = e ^ { x }$ . State the domain, range, and asymptote. $h ( x ) = e ^ { x - 3 } + 5$

Question 38

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

Question 39

Find the exact solution of the equation. $e ^ { 3 x ^ { 4 } } = 10$

Question 40

Solve the inequality: $0.5 \leq \log x < 2$

Question 41

How much more intense is an earthquake with a magnitude of $9.5$
on the Richter scale than one with a magnitude of $7.6$ ?

Question 42

Evaluate the expression without using a calculator.

A)
B)
C)
D)
E)

Question 43

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If $D _ { 0 }$ is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by $A = - 8267 \ln \left( \frac { D } { D _ { 0 } } \right)$ Find the age of an object if the amount D of carbon-14 that remains in the object is 85% of the original amount $D _ { 0 }$
.

Question 44

A substance has a hydrogen ion concentration of $\left[ \mathrm { H } ^ { + } \right] = 4.0 \times 10 ^ { - 3 }$
M. Find the pH of the substance.

Question 45

Use a graphing device to find the solutions of the equation, correct to two decimal places. $9 - x ^ { 2 } = e ^ { - 3 x }$

Question 46

Expand the logarithmic expression.

A)
B)
C)
D)
E) none

Question 47

Sketch the graph of the function.

A)

B)
C)

D)

Question 48

How long will it take an investment of $\$ 300$
to triple, if the interest rate is $9.5 \%$
per year compounded continuously?

Question 49

Find the domain, range, and asymptote of the function.

A) Domain:
, Range:
, Asymptote:
B) Domain:
, Range:
, Asymptote:
C) Domain:
, Range:
, Asymptote:
D) Domain:
, Range:
, Asymptote:
E) none of these

Question 50

Radium - $226$ has a half-life of $1600 \text { years }$ . Suppose we have a $300 \mathrm {~g}$ sample.
(a) Find a formula for the mass remaining after ${ t }$ years
b) How much of the sample remains after $200$ years?
c) After how long will only $20 \mathrm {~g}$ remain?

Question 51

Evaluate the expression without using a calculator.

A)
B)
C)
D)
E) none of these

Question 52

Write the equation in logarithmic form.

A)
B)
C)
D)
E)

Question 53

Find the decibel intensity level at a crowded football game if the intensity was measured at $3.98 \times 10 ^ { - 3 }$ $\text{ watts/m}^2$

Question 54

Find the domain of the function.

A)
B)
C)
D)
E) none of these

Question 55

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms.

A)
B)
C)
D)
E)

Question 56

Expand the logarithmic expression.

A)
B)
C)
D)
E)

Question 57

The velocity of a sky diver t seconds after jumping is given by
. After how many seconds is the velocity 60 ft/s?

Question 58

Evaluate the expression without using a calculator.

A)
B)
C)
D)
E)

Question 59

How much more intense is an earthquake with a magnitude of $9.5$ on the Richter scale than one with a magnitude of $7.6$ ?

A) $1.9$ Times
B) $79$ Times
C) $2.9$ Times
D) $150$ Times
E) none of these

Question 60

Radium - $226$ has a half-life of $1600 \text { years }$ ) Suppose we have a $300 \mathrm {~g}$
Sample. How much of the sample remains after $200$ Years?

A) $105.61 \mathrm {~g}$
B) $124.61 \text { g }$
C) $275.1 \mathrm {~g}$
D) $288.1 \mathrm {~g}$
E) $200.1 \mathrm {~g}$

Question 61

A substance has a hydrogen ion concentration of $\left[ \mathrm { H } ^ {+ } \right] = 4.0 \times 10 ^ { - 3 }$ M. Find the pH of the substance

A)pH $\approx 7.4$
B)pH $\approx 5.4$
C) pH $\approx 3.4$
D)pH $\approx 4.4$
E)pH $\approx 2.4$

Question 62

Sketch the graph of the function.

A)

B)
C)

D)

Question 63

Solve the inequality:

A)
B)
C)
D)
E)

Question 64

Solve the equation. Find the exact solution. $e^{3 x / 5}=7$

A) $x = \frac { 5 } { 3 } \ln 7$

B) $x = \frac { 7 } { 3 }$

C) $x = \frac { 7 } { 3 } \ln 5$

D) $x = 7 \ln ( 5 / 3 )$

E) $x = \frac { e } { 7 } \ln 3$

Question 65

Write the equation in logarithmic form. $25 ^ { - 1 / 2 } = \frac { 1 } { 5 }$

A) $\log _ { 5 } \left( \frac { 1 } { 2 } \right) = 25$

B) $\log _ { 25 } \left( \frac { 1 } { 5 } \right) = - \frac { 1 } { 2 }$

C) $\log _ { 5 } \left( \frac { 1 } { 5 } \right) = - \frac { 1 } { 25 }$

D) $\log _ { 5 } \left( \frac { 1 } { 5 } \right) = - \frac { 1 } { 2 }$

E) none

Question 66

Approximately how long will it take an investment of $\$ 300$ to triple, if the interest rate is $3.5 \%$ Per year compounded continuously?

A) $3.5 \text { years }$
B) $21 \text { years }$
C) $13 \text { years }$
D) $31 \text { years }$
E) $9 \text { years }$

Question 67

Find the domain, range, and asymptote if any. $h ( x ) = e ^ { x + 1 } - 5$

A)Domain: $( - \infty , \infty )$
; Range: $( 5 , \infty )$
; Asymptote: $y = - 5$

B)Domain: $( - \infty , \infty )$
; Range: $( - 5 , \infty )$
; Asymptote: $y = 5$

C)Domain: $( - 5 , \infty )$
; Range: $( - 5 , \infty )$
; Asymptote: $y = - 5$

D)Domain: $( - \infty , \infty )$
; Range: $( - \infty , \infty )$
; Asymptote: $\text { no asymptote }$

E) none of these

Question 68

Combine into a single logarithm. $\log ( x - 3 ) + \log ( x + 3 ) - \frac { 1 } { 2 } \log \left( x ^ { 2 } + 9 \right)$

A) $\log \left( \frac { x ^ { 2 } - 9 } { x ^ { 2 } + 9 } \right)$

B) $\log \left( \frac { \sqrt { x ^ { 2 } - 9 } } { x ^ { 2 } } + 9 \right)$

C) $\log \left( \frac { x ^ { 2 } - 9 } { \sqrt { x ^ { 2 } + 9 } } \right)$

D) $\log \left( \frac { x - 3 } { x + 3 } \right)$

E) $\log \left( \frac { x - 9 } { \sqrt { x + 9 } } \right)$

Question 69

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If $D _ { 0 }$ is the original amount of carbon-14 and D is the amount remaining, then the artifact's age A (in years) is given by $A = - 8267 \ln \left( \frac { D } { D _ { 0 } } \right)$ Find the age of an object if the amount D of carbon-14 that remains in the object is 50% of the original amount $D _ { 0 }$

A) $4134 \mathrm { yr }$
B) $5730 \mathrm { yr }$
C) $2067 \mathrm { yr }$
D) $16400 \mathrm { yr }$
E) none of these

Question 70

Find the decibel intensity level at a rock concert if the intensity was measured at $3.98 \times 10 ^ { - 3 }$ $\text {watts/m }^2$

A) $12 \mathrm {~dB}$
B) $114 \mathrm {~dB}$
C) $52 \mathrm {~dB}$
D) $88 \mathrm {~dB}$
E) none of these

Question 71

Combine into a single logarithm. $\frac { 3 } { 2 } \ln ( x - y ) - 2 \ln \left( x ^ { 2 } + y ^ { 2 } \right)$

A) $\ln \frac { ( x - y ) ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$

B) $\ln \frac { \sqrt [ 3 ] { ( x - y ) ^ { 2 } } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } }$

C) $\ln \frac { \sqrt { ( x - y ) ^ { 3 } } } { \left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } }$

D) $\ln \frac { \sqrt [ 3 ] { ( x - y ) ^ { 2 } } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$

E) $\ln \frac { \sqrt { ( x - y ) ^ { 3 } } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$

Question 72

A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function $m ( t ) = 13 e ^ { - 0.015 t }$ where $m ( \text { t })$ Is measured in kilograms. Round your answers to three decimal places. Find the mass at time $t = 0$

A) $13.5 \mathrm {~kg}$
B) $0.015 \mathrm {~kg}$
C) $13.015 \mathrm {~kg}$
D) $15.135 \mathrm {~kg}$
E)none of these

Question 73

Solve the equation. Find the exact solution.

A)
B)
C)
D)
E)

Question 74

Use a graphing device to find the solutions of the equation, correct to two decimal places.

A)
B)
C)
D)
E)

Question 75

Loren wants to invest $\$ 6,500$ in a fund at an interest rate of $7.5 \%$ Per year, compounded monthly. How long will it take her investment to grow to $\$ 7,850 ?$

A) $2.5 \text { years }$
B) $25 \text { years }$
C) $0.5 \text { years }$
D) $4.5 \text { years }$
E) $1 \text { year }$

Question 76

Find the domain, range, and asymptote of the function. $g ( x ) = \log _ { 2 } ( x + 4 )$

A) Domain: $( - 4 , \infty )$
, Range: $( - 4 , \infty )$
, Asymptote: $x = - 4$

B) Domain: $( 4 , \infty )$
, Range: $( 4 , \infty )$
, Asymptote: $x = 4$

C) Domain: $( - 4 , \infty )$
, Range: $(-\infty, \infty )$
, Asymptote: $x = - 4$

D) Domain: $( - 4 , \infty )$
, Range: $( - 4 , \infty )$
, Asymptote: $x = 4$

E) none of these

Question 77

Solve the equation. Find the exact solution.

A)
B)
C)
D)
E)

Question 78

Evaluate the expression without using a calculator.

A)
B)
C)
D)
E) none

Question 79

The velocity of a sky diver t seconds after jumping is given by $v ( t ) = 80 \left( 1 - e ^ { - 02 t } \right)$ . After how many seconds is the velocity 50 ft/s?

A)5.0 s
B)4.9 s
C)8.9 s
D)0.2 s
E)none of these

Question 80

Brian wants to invest $\$ 6,500$ at an interest rate of $7.5 \%$ Per year, compounded daily. How long will it take his investment to grow to $\$ 8,000 ?$

A) $2.7 \text { years }$
B) $27 \text { years }$
C) $0.27 \text { years }$
D) $37 \text { years }$
E) $1 \text { year }$