AnswersMathematics IIIMathematical Modeling: Part Two

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A biologist begins working with a sample containing 20 comma 000 bacteria. The population size doubles every 10 days according to the equation p times t is equal to 20 comma 000 times 2 raised to the t over 10 power , where t represents the number of days. After seven days the biologist begins working with a second sample of 20 comma 000 bacteria. The equation used to represent the population of sample 2 after t days is p times t is equal to 20 comma 000 times 2 raised to the the fraction with numerator t minus 7 and denominator 10 power Which of the following is an equivalent form of the equation for the population of sample 2 ?

A
p times t is equal to the fraction with numerator 20 comma 000 times 2 raised to the t minus 7 power and denominator 2 to the 10th power
B
p times t is equal to the fraction with numerator 20 comma 000 10th root of 2 to the t th power and denominator 10th root of 2 to the 7th power
C
p times t is equal to the fraction with numerator 20 comma 000 times 7th root of 2 to the t th power and denominator 2 to the 10th power
D
p times t is equal to the fraction with numerator 20 comma 000 times 2 raised to the t over 10 power and denominator 2 to the 7th power
2
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Which expressions are equivalent to open paren 2 to the 5th power close paren to the negative 2 power ?

A
2 to the negative 10 power and 1 over 20
B
10 to the negative 2 power and 1 over 100
C
2 to the negative 10 power and 1 over 1024
D
10 to the negative 10 power and 1 over 100
3

What are the domain and range of f of x is equal to the log of open paren x minus 1 close paren plus 2 ?

A
domain : all real numbers; range: y is greater than 1
B
domain: x is greater than 1 ; range: all real numbers
C
domain: all real numbers; range: all real numbers
D
domain: x is greater than 1 ; range: y is greater than 2
4

One brand of vinegar has a pH of 4 point 5 Another brand has a pH of 5 point 0 The equation for the pH of a substance is pH is equal to negative log times open bracket cap h raised to the plus power close bracket , where H+ is the concentration of hydrogen ions. What is the approximate difference in the concentration of hydrogen ions between the two brands of vinegar?

A
2 point 2 times 10 to the negative 5 power
B
6 point 8 times 10 to the 4th power
C
3 point 2 times 10 inverse
D
3 point 2 times 10 to the 1th power
5

What is the solution of the log base x of 729 is equal to 3 ?

A
9
B
81
C
18
D
6
6

The two expressions have the same value when rounded to the nearest hundredth. the log base 5 of b the log base 9 of 48 What is the approximate value of log b to the nearest hundredth?

A
1 point 2 3
B
0 point 9 3
C
9 point 1 6
D
65 point 5 3
7

Which expression has the same value as negative y to the negative 4 power ?

A
negative the fraction with numerator 1 and denominator y to the 4th power
B
the fraction with numerator 1 and denominator y to the 4th power
C
negative y to the 4th power
D
y to the 4th power
8

What is the range of the function f of x is equal to negative 2 times open paren 6 to the x th power close paren plus 3 ?

A
left bracket 3 comma infinity right parenthesis
B
left parenthesis negative infinity comma negative 2 right bracket
C
left bracket negative 2 comma infinity right parenthesis
D
open paren negative infinity comma 3 close paren
9

Which of the following shows the graph of y is equal to the natural log of open paren negative 2 x close paren ?

A
Image with description On a coordinate plane, the x and y axis ranges from negative 5 to 5 with increment of 1. There is a curved graph that starts from the quadrant I, crosses the x axis at point 4 comma 0, decreases with x value. It approaches y axis in the quadrant IV.
B
Image with description On a coordinate plane, the x and y axis ranges from negative 5 to 5 with increment of 1. There is a curved graph that starts from the quadrant II, crosses the x axis at 0 point 5 comma 0, decreases with the increased value of x. It approaches y axis in the quadrant III.
C
Image with description On a coordinate plane, the x and y axis ranges from negative 5 to 5 with increment of 1. There is a curved graph that starts from the quadrant II, crosses the x axis at point negative 4 comma 0, decreases with the increased value of x. It approaches y axis in the quadrant III.
D
Image with description On a coordinate plane, the x and y axis ranges from negative 5 to 5 with increment of 1. There is a curved graph that starts from the quadrant I, crosses the x axis at 0 point 5 comma 0, decreases with x value. It approaches y axis in the quadrant IV.
11

What is log sub 6 e to the f th power rewritten using the power property?

A
f log sub 6 e
B
log sub 6 of open paren e plus f close paren
C
log sub 6 of open paren e times f close paren
D
e log sub 6 f
12

What are the solutions to log sub 6 of open paren x squared plus 8 close paren is equal to 1 plus log sub 6 of x ?

A
x is equal to negative 2 and x is equal to negative 4
B
x is equal to 2 and x is equal to 4
C
x is equal to negative 1 and x is equal to 8
D
x is equal to 1 and x is equal to negative 8
13

Two different radioactive isotopes decay to 10 percent of their respective original amounts. Isotope A does this in 33 days, while isotope B does this in 43 days. What is the approximate difference in the half-lives of the isotopes?

A
13 days
B
10 days
C
3 days
D
33 days
14

For what value of x does 625 is equal to 5 raised to the 6 minus 2 x power ?

A
5
B
4
C
2
D
1
15

Which graph shows the solution to the equation? log sub 3 of open paren x plus 2 close paren is equal to 1

A
Image with description Coordinate plane with x-axis and y-axis. A logarithmic-shaped curve begins near the y-axis at x equals 0 and extends to the right. The curve passes through 1 comma 0, rises gradually through approximately 3 comma 1 and 9 comma 2, and continues increasing as x increases. A horizontal line y equals negative 1 extends across the graph with arrows on both ends. The curve intersects the horizontal line near 0.3 comma negative 1 and lies above the line for x values greater than this point. Both the curve and the horizontal line continue indefinitely as indicated by arrows.
B
Image with description On a coordinate plane the x axis lists from negative 3 to 9 by 1 and y axis lists from negative 4 to 8 by 1. There is a smooth curve that approaches the line x equals negative 2 in the third quadrant. It passes through the point (negative 1 comma 0), (8 comma 1) and increases as x increases. There is a line that is parallel to the x axis and cuts the y axis at (0 comma 1).
C
Image with description On a coordinate plane the x axis lists from negative 3 to 9 by 1 and y axis lists from negative 4 to 8 by 1. There is a smooth curve that approaches the line x equals negative 2 in the third quadrant. It passes through the point (negative 1 comma 0), (8 comma 1) and increases as x increases. There is a line that is parallel to the x axis and cuts the y axis at (0 comma negative 1).
D
Image with description On a coordinate plane the x axis lists from negative 3 to 9 by 1 and y axis lists from negative 4 to 8 by 1. There is a smooth curve that approaches the line x equals negative 2 in the third quadrant. It passes through the point (negative 1 comma 0), (1 comma 1) and increases as x increases. There is a line parallel to the x axis and cuts the y axis at (0 comma 1).
16

Which function has a horizontal asymptote of y is equal to 3 ?

A
f of x is equal to 2 times 4 raised to the x minus 3 power
B
f of x is equal to 2 times open paren 3 to the x th power close paren
C
f of x is equal to 2 times open paren 4 to the x th power close paren plus 3
D
f of x is equal to 3 times open paren 2 to the x th power close paren
17

What is the solution of the natural log of open paren x plus 6 close paren minus natural log 9 is equal to 2 ? Round your answer to the nearest hundredth.

A
x is equal to 60 point 5 0
B
x is equal to 66 point 5 0
C
x is equal to 72 point 5 0
D
x is equal to 53 point 5 0
18

Which of the following shows the true solution to the logarithmic equation solved? log sub 2 of x plus log sub 2 of open paren x plus 7 close paren is equal to 3 log sub 2 of open bracket x times open paren x plus 7 close paren close bracket is equal to 3 x times open paren x plus 7 close paren is equal to 2 cubed x squared plus 7 x minus 8 is equal to 0 open paren x plus 8 close paren times open paren x minus 1 close paren is equal to 0 x is equal to negative 8 comma 1

A
x is equal to negative 8
B
x is equal to 1
C
x is equal to 1 and x is equal to 8
D
x is equal to 1 and x is equal to negative 8
19

What is the domain of the function y is equal to the natural log of open paren x plus 2 close paren ?

A
x is greater than negative 2
B
x is less than negative 2
C
x is greater than 2
D
x is less than 2
20

Which expression is equivalent to log sub 2 9 x cubed ?

A
the log base 2 of x plus 3 log sub 2 9
B
3 log sub 2 x minus the log base 2 of 9
C
the log base 2 of 9 plus 3 log sub 2 x
D
3 log sub 2 9 minus the log base 2 of x
21

The intensity, or loudness, of a sound can be measured in decibels d cap b , according to the equation cap i times d cap b is equal to 10 log open bracket the fraction with numerator cap i and denominator cap i sub 0 close bracket , where cap i is the intensity of a given sound and cap i sub 0 is the threshold of hearing intensity. What is the intensity, in decibels, open bracket cap i times d cap b close bracket , when cap i is equal to 10 to the 32th power times open paren cap i sub 0 close paren ?

A
737
B
320
C
32
D
15

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