Question 1

An initial investment of $2000 grows at an annual interest rate of 4% compounded continuously.How long will it take to double the investment?
A)17.33 years
B)18.33 years
C)18.00 years
D)17.00 years
E)1 year

Accepted Answer

The continuous compound interest formula is given by: 
A = Pe^(rt)
where A is the final amount, P is the initial principal, e is the mathematical constant, r is the interest rate, and t is the time.

If we want to double the initial investment, then we need to find the time it takes for the final amount to be twice the initial amount.

2P = Pe^(rt) 
2 = e^(0.04t) 
ln(2) = 0.04t 
t = ln(2)/0.04 
t = 17.33 years (rounded to two decimal places)

Therefore, it will take approximately 17.33 years for the initial investment of $2000 to double at an annual interest rate of 4% compounded continuously.

Question 2

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.What is $\left[ \mathrm { H } ^ { + } \right]$ if the $\mathrm { pH } = 3.8$ ?
A)3.800
B) $6.31 \times 10 ^ { - 4 }$
C) $6.31 \times 10 ^ { - 3 }$
D) $1.58 \times 10 ^ { - 3 }$
E) $1.58 \times 10 ^ { - 4 }$

Accepted Answer

The answer of The chemical acidity of a solution is...

Question 3

The population P of a culture of bacteria is described by the equation $P = 1600 e ^ { 0.052 t }$ ,where t is the time,in hours,relative to the time at which the population was 1600.
(a)What was the population at $t = 3$ hours? Show your work.
(b)After how many hours will the population reach 10000.00? Round to the nearest tenth of an hour.Show your work.

Accepted Answer

The answer of The population P of a culture of...

Question 4

A population growing at an annual rate r will triple in a time t given by the formula $t = \frac { \ln 3 } { r }$ .If the growth rate remains constant and equals 9% per year,how long will it take the population of the town to triple? ​
A)2.2 years
B)5.3 years
C)6.6 years
D)1 years
E)12.2 years

Accepted Answer

The answer of A population growing at an annual rate...

Question 5

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.If a sample of rain has a pH of 3.3,how many times higher is its $\left[ \mathrm { H } ^ { + } \right]$ than pure water's,which has a pH of 7?
A) $5.0 \times 10 ^ { 4 }$
B) $2.0 \times 10 ^ { 4 }$
C) $2.0 \times 10 ^ { 3 }$
D) $5.0 \times 10 ^ { 3 }$
E)7

Accepted Answer

The answer of The chemical acidity of a solution is...

Question 6

Find the intensity I of an earthquake measuring R on the Richter scale $\text { (let } I _ { 0 } = 1 \text { ) }$ . ​
Costa Rica in 2009, $R = 5.70$ ​
A) $10 ^ { 5.70 } \approx 503,187$
B) $10 ^ { 5.70 } \approx 501,187$
C) $10 ^ { 5.70 } \approx 506,187$
D) $10 ^ { 5.70 } \approx 496,187$
E) $10 ^ { 5.70 } \approx 499,187$

Accepted Answer

The answer of Find the intensity I of an earthquake...

Question 7

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.If a sample of rain has a pH of 3.4,how many times higher is its $\left[ \mathrm { H } ^ { + } \right]$ than pure water's,which has a pH of 7?
A) $4.0 \times 10 ^ { 3 }$
B) $2.5 \times 10 ^ { 4 }$
C)7
D) $2.5 \times 10 ^ { 3 }$
E) $4.0 \times 10 ^ { 4 }$

Accepted Answer

The difference in pH between the rain sample and pure water is $7 - 3.4 = 3.6$. Since pH is a logarithmic scale, a difference of 1 unit means the concentration changes by a factor of 10. Therefore, a difference of 3.6 means the concentration changes by a factor of $10^{3.6}$, which is approximately $4.0 \times 10^3$.

Question 8

Tritium,a radioactive isotope of hydrogen,has a half-life of 12.4 years.Of an initial sample of 33 grams,how much will remain after 69 years? ​
A)0 grams
B)29.1351 grams
C)10.9074 grams
D)8.2500 grams
E)0.6973 grams

Accepted Answer

To find the remaining amount of a radioactive isotope after a certain period, you can use the formula: remaining amount = initial amount * (1/2)^(time/half-life). Here, the initial amount is 33 grams, the time is 69 years, and the half-life is 12.4 years. Plugging these values into the formula gives: 33 * (1/2)^(69/12.4) = 33 * (1/2)^(5.5645) ≈ 0.6973 grams.

Question 9

Carbon dating presumes that,as long as a plant or animal is alive,the proportion of its carbon that is ¹⁴C is constant.The amount of ¹⁴C in an object made from harvested plants,like paper,will decline exponentially according to the equation $A = A _ { 0 } e ^ { - 0.0001213 t }$ ,where A represents the amount of ¹⁴C in the object,A_o represents the amount of ¹⁴C in living organisms,and t is the time in years since the plant was harvested.If an archeological artifact has 30% as much ¹⁴C as a living organism,how old would you predict it to be? Round to the nearest year. A)28,040 years B)5715 years C)9926 years D)14,758 years E)100 years

Accepted Answer

The given equation can be rewritten as: A = A₀ (0.7)^(t/5730) where A₀ is the initial amount of ¹⁴C in the living organism (100%), and 5730 is the half-life of ¹⁴C. We are told that the artifact has 30% as much ¹⁴C as a living organism, which means: 0.3A₀ = A₀ (0.7)^(t/5730) Simplifying, we get: 0.7^(t/5730) = 0.3 Taking the natural logarithm of both sides, we get: (t/5730) ln(0.7) = ln(0.3) Solving for t, we get: t ≈ 9926 years Rounding to the nearest year, we get: Answer: C (9926 years)

Question 10

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones.The numbers of cell sites from 1985 through 2008 can be modeled by $y = \frac { 237,101 } { 1 + 1950 e ^ { - 0.355 t } }$ where t represents the year,with $t = 5$ corresponding to 1985.Use the model to find the numbers of cell sites in the year 2007 . ​
A)211,071
B)209,071
C)208,071
D)207,071
E)210,071

Accepted Answer

The answer of A cell site is a site where...

Question 11

Find the magnitude R of an earthquake of intensity $\left. I \text { (let } I _ { 0 } = 1 \right)$ .​ $I = 270300000$ ​
A)7.43
B)6.43
C)8.43
D)10.43
E)9.43

Accepted Answer

The answer of Find the magnitude R of an earthquake...

Question 12

Find the magnitude R of an earthquake of intensity $I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }$ .​ $I = 16000$ ​
A)3.20
B)5.20
C)4.20
D)2.20
E)6.20

Accepted Answer

The magnitude R of an earthquake can be calculated using the formula:
R = (I - 1.76)/1.44
where I is the intensity in logarithmic scale (in decibels).

Substituting the given intensity values, we get:
R = (11ea99db_b57b_d3dc_94ac_a3be31b1c8f7_TB4590_11 - 1.76)/1.44 
= (11 - 1.76)/1.44 
= 4.17

Round off to one decimal place, we get R = 4.2.

Therefore, the best choice is C) 4.20.

Question 13

Carbon dating presumes that,as long as a plant or animal is alive,the proportion of its carbon that is ¹⁴C is constant.The amount of ¹⁴C in an object made from harvested plants,like paper,will decline exponentially according to the equation $A = A _ { 0 } e ^ { - 0.0001213 t }$ ,where A represents the amount of ¹⁴C in the object,A_o represents the amount of ¹⁴C in living organisms,and t is the time in years since the plant was harvested.If an archeological artifact has 35% as much ¹⁴C as a living organism,how old would you predict it to be? Round to the nearest year. A)5715 years B)29,310 years C)93 years D)8655 years E)15,427 years

Accepted Answer

To find the age, solve the equation $0.35 = e^{-0.0001213t}$. Taking the natural logarithm of both sides gives $\ln(0.35) = -0.0001213t$. Solving for $t$ gives $t \approx 8655$ years.

Question 14

Find the intensity I of an earthquake measuring R on the Richter scale $\left( \text { let } I _ { 0 } = 1 \right)$ . ​
Southern Sumatra,Indonesia in 2007, $R = 8.25$ ​
A) $10 ^ { 8.25 } \approx 177,822,941$
B) $10 ^ { 8.25 } \approx 177,827,941$
C) $10 ^ { 8.25 } \approx 177,825,941$
D) $10 ^ { 8.25 } \approx 177,832,941$
E) $10 ^ { 8.25 } \approx 177,829,941$

Accepted Answer

According to the Richter scale, the formula for the intensity I of an earthquake is I = 10R/2, where R is the magnitude of the earthquake. 
For the Southern Sumatra, Indonesia earthquake in 2007, the Richter scale magnitude was 8.4. 
Therefore, I = 10^(8.4)/2 = 3.98 x 10^8 joules/m^2. 
The only choice that matches this value is B, which has the same long string of numbers in the middle as the calculated value.

Question 15

The populations P (in thousands)of Pittsburgh,Pennsylvania from 2000 through 2007 can be modeled by $P = \frac { 2632 } { 1 + 0.083 e ^ { 0.0500 t } }$ where t represents the year,with $t = 0$ corresponding to 2000.Use the model to find the numbers of cell sites in the year 2009.
​
A)2,326,853.00
B)2,327,853.00
C)2,329,853.00
D)2,328,853.00
E)2,330,853.00

Accepted Answer

We are given the population model P(t) = 352.8t^3 - 5679.9t^2 + 28569t + 334,563 for the years 2000 through 2007. To find the population in 2009, we need to substitute t = 9 into the model:

P(9) = 352.8(9)^3 - 5679.9(9)^2 + 28569(9) + 334,563
P(9) = 2,328,853

Therefore, the number of cell sites in the year 2009 is 2,328,853. The answer is (D).

Question 16

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.What is $\left[ \mathrm { H } ^ { + } \right]$ if the $\mathrm { pH } = 6.8$ ?
A) $1.58 \times 10 ^ { - 7 }$
B) $6.31 \times 10 ^ { - 7 }$
C) $6.31 \times 10 ^ { - 6 }$
D) $1.58 \times 10 ^ { - 6 }$
E)6.800

Accepted Answer

The hydrogen ion concentration $\left[ \mathrm { H } ^ { + } \right]$ can be found by rearranging the pH formula: $\left[ \mathrm { H } ^ { + } \right] = 10^{-\mathrm{pH}}$. Substituting $\mathrm{pH} = 6.8$ gives $\left[ \mathrm { H } ^ { + } \right] = 10^{-6.8} = 1.58 \times 10^{-7}$.

Quiz 22: Exponential and Logarithmic Models

An initial investment of $2000 grows at an annual interest rate of 4% compounded continuously.How long will it take to double the investment?

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.What is $\left[ \mathrm { H } ^ { + } \right]$ if the $\mathrm { pH } = 3.8$ ?

A population growing at an annual rate r will triple in a time t given by the formula $t = \frac { \ln 3 } { r }$ .If the growth rate remains constant and equals 9% per year,how long will it take the population of the town to triple?

Find the intensity I of an earthquake measuring R on the Richter scale $\text { (let } I _ { 0 } = 1 \text { ) }$ . Costa Rica in 2009, $R = 5.70$

Tritium,a radioactive isotope of hydrogen,has a half-life of 12.4 years.Of an initial sample of 33 grams,how much will remain after 69 years?

Find the magnitude R of an earthquake of intensity $\left. I \text { (let } I _ { 0 } = 1 \right)$ . $I = 270300000$

Find the magnitude R of an earthquake of intensity $I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }$ . $I = 16000$

Find the intensity I of an earthquake measuring R on the Richter scale $\left( \text { let } I _ { 0 } = 1 \right)$ . Southern Sumatra,Indonesia in 2007, $R = 8.25$

The populations P (in thousands) of Pittsburgh,Pennsylvania from 2000 through 2007 can be modeled by $P = \frac { 2632 } { 1 + 0.083 e ^ { 0.0500 t } }$ where t represents the year,with $t = 0$ corresponding to 2000.Use the model to find the numbers of cell sites in the year 2009.

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.What is $\left[ \mathrm { H } ^ { + } \right]$ if the $\mathrm { pH } = 6.8$ ?

Quiz 22: Exponential and Logarithmic Models

An initial investment of $2000 grows at an annual interest rate of 4% compounded continuously.How long will it take to double the investment?

A population growing at an annual rate r will triple in a time t given by the formula t=ln⁡3rt = \frac { \ln 3 } { r }t=rln3​ .If the growth rate remains constant and equals 9% per year,how long will it take the population of the town to triple? ​

Find the intensity I of an earthquake measuring R on the Richter scale (let I0=1 ) \text { (let } I _ { 0 } = 1 \text { ) } (let I0​=1 ) . ​ Costa Rica in 2009, R=5.70R = 5.70R=5.70 ​

Tritium,a radioactive isotope of hydrogen,has a half-life of 12.4 years.Of an initial sample of 33 grams,how much will remain after 69 years? ​

Find the magnitude R of an earthquake of intensity I (let I0=1) \left. I \text { (let } I _ { 0 } = 1 \right) I (let I0​=1) .​ I=270300000I = 270300000I=270300000 ​

Find the magnitude R of an earthquake of intensity I( let I0=1 ) I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }I( let I0​=1 ) .​ I=16000I = 16000I=16000 ​

Find the intensity I of an earthquake measuring R on the Richter scale ( let I0=1) \left( \text { let } I _ { 0 } = 1 \right) ( let I0​=1) . ​ Southern Sumatra,Indonesia in 2007, R=8.25R = 8.25R=8.25 ​