Question 1

Soon after taking an aspirin, a patient has absorbed 350 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially, with half being removed every 2 hours, find the time it will take for the amount of aspirin in the bloodstream to decrease to 290 mg. Select the answer rounded to three decimal places.

Accepted Answer

The exponential decay formula is $A = P(0.5)^{\frac{t}{h}}$, where $A$ is the amount after time $t$, $P$ is the initial amount, $0.5$ represents the half-life decay rate, and $h$ is the half-life in hours. Here, $P = 350$ mg, $A = 290$ mg, and $h = 2$ hours. Plugging these values into the formula gives $290 = 350(0.5)^{\frac{t}{2}}$. Solving for $t$ gives approximately $1.843$ hours.

Question 2

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 14 grams of plutonium-239 to decay to 1 grams? Select the correct answer rounded to the nearest integer.

Accepted Answer

To solve this problem we use the formula for radioactive decay: 
N(t) = N0 * (1/2)^(t/T) 
Where N(t) is the amount remaining after time t, N0 is the initial amount, T is the half-life, and (1/2)^(t/T) is the fraction remaining after time t. 
We are given N0 = 14 grams, N(t) = 1 gram, and T = 24,400 years. 
Substituting these values we get: 
1 = 14 * (1/2)^(t/24,400) 
(1/2)^(t/24,400) = 1/14 
Taking the natural logarithm of both sides: 
ln[(1/2)^(t/24,400)] = ln(1/14) 
(t/24,400) * ln(1/2) = ln(1/14) 
t = 24,400 * ln(1/14) / ln(1/2) 
Using a calculator we get t ≈ 92,900 years. 
Therefore, the best choice is C, rounded to the nearest integer.

Question 3

The given table corresponds to the function  .

Accepted Answer

The answer of The given table corresponds to the function...

Question 4

Use logarithms to solve the equation. (Round answers to four decimal places.)

Accepted Answer

The answer of Use logarithms to solve the equation. (Round...

Question 5

Use logarithms to solve the equation. (Round answers to four decimal places.)  x = __________

Accepted Answer

The answer of Use logarithms to solve the equation. (Round...

Question 6

Graph the function.

Accepted Answer

The answer of Graph the function. 
...

Question 7

How long will it take an investment to triple if it is continuously compounded at 15% per year? Select the correct answer rounded to the nearest year.

Accepted Answer

The formula for continuous compounding is: 
A = Pe^(rt) 
where A is the amount, P is the principal (initial investment), r is the annual interest rate (as a decimal), and t is the time (in years).

We want to find the time it takes for the investment to triple, so we need to solve for t: 
3P = Pe^(0.15t) 
Divide both sides by P: 
3 = e^(0.15t)

Take the natural logarithm of both sides: 
ln(3) = 0.15t 
t = ln(3)/0.15

Using a calculator, we get t ≈ 7.27 years. Rounded to the nearest year, the answer is 7 years. Therefore, the best choice is C.

Question 8

The given table corresponds to the function  .

Accepted Answer

The given table corresponds to two different functions, 11ebfe6a_2cb0_930d_96cc_45e3a4051bb2_TBX8650_11 and 11ebfe6a_2cb0_930e_96cc_f7a2871b0d5c_TBX8650_00. Therefore, the answer is A, which means the statement is true.

Question 9

You are trying to determine the half-life of a new radioactive element you have isolated. You start with 3 grams, and 4 days later you determine that it has decayed down to 0.1 gram. What is the half-life? Round your answer to three decimal places. Select the answer rounded to three decimal places.

Accepted Answer

The half-life of a radioactive element is the amount of time it takes for half of the element to decay. In this case, we start with 3 grams of the element, and 4 days later we determine that it has decayed down to 0.1 gram. This means that half of the element has decayed in 4 days, so the half-life is 4 days.

Question 10

Graph the function.

Accepted Answer

The answer of Graph the function. 
...

Question 11

How long, to the nearest year, will it take an investment in Japan to double its value if the interest is compounded every six months? Please round the answer to the nearest year. Country U.S. Japan Canada Korea Australia Yield 5.3% 1.5% 5.2% 5.4% 6.0%

Accepted Answer

The formula for compound interest is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times per year that the interest is compounded, and t is the number of years. In this case, we have P = 1, r = 1.5%, n = 2, and we want to find t such that A = 2. Plugging these values into the formula, we get 2 = 1(1 + 0.015/2)^(2t). Solving for t, we get t = 46.38 years. Rounding to the nearest year, we get 46 years.

Question 12

The table lists interest rates on long-term investments (based on 10-year government bonds)in several countries in 2004-2005. Assuming that you invest $11,000 in Japan, how long (to the nearest year)must you wait before your investment is worth $17,000 if the interest is compounded annually? Country U.S. Japan Canada Yield 5.3% 1.5% 5.2%

Accepted Answer

To find the number of years required for an investment to grow from $11,000 to $17,000 with an annual interest rate of 1.5% (Japan's rate), we use the formula for compound interest: $A = P(1 + r)^t$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount ($11,000), $r$ is the annual interest rate (1.5% or 0.015), and $t$ is the time the money is invested for in years.Rearranging the formula to solve for $t$, we get:$$17000 = 11000(1 + 0.015)^t$$Dividing both sides by 11000:$$1.54545 = (1.015)^t$$Taking the natural logarithm of both sides to solve for $t$:$$ln(1.54545) = t \cdot ln(1.015)$$$$t = \frac{ln(1.54545)}{ln(1.015)}$$$$t ≈ 29$$Therefore, it takes approximately 29 years for the investment to grow to $17,000.

Question 13

Find the associated exponential decay model.  when  ; Half-life

Accepted Answer

The answer of Find the associated exponential decay model. ...

Question 14

Find the associated doubling time.

Accepted Answer

The answer of Find the associated doubling time. 
...

Question 15

How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 20% per year? Round the answers to the nearest year. __________ years

Accepted Answer

The answer of How long, to the nearest year, will...

Question 16

The given table corresponds to the function  .

Accepted Answer

The given table corresponds to a function, indicated by the long alphanumeric string in the question. The fact that there are two different strings for the function (ending in TBX8650_11 and TBX8650_00) indicates that there may be different versions or iterations of the same function. However, based on the information provided, it is true that the table corresponds to the function represented by those strings.

Question 17

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.

Accepted Answer

There are too many unknowns in the given expression to determine the correct answer without additional information.

Question 18

The amount of carbon-14 remaining in a sample that weighs A is given by  where t is time in years. If tests on a fossilized skull reveal that 99.97% of the carbon-14 has decayed, how old is the skull? Select the correct answer rounded to the nearest integer.

Accepted Answer

The answer of The amount of carbon-14 remaining in a...

Question 19

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.

Accepted Answer

The answer of Convert the exponential function to the form...

Question 20

Use logarithms to solve the equation. (Round answers to four decimal places.)

Accepted Answer

The answer of Use logarithms to solve the equation. (Round...

Quiz 2: Nonlinear Functions and Models

Plutonium-239 is used as a fuel for some nuclear reactors and also as the fissionable material in atomic bombs. It has a half-life of 24,400 years. How long will it take 14 grams of plutonium-239 to decay to 1 grams? Select the correct answer rounded to the nearest integer.

The given table corresponds to the function .

Use logarithms to solve the equation. (Round answers to four decimal places.)

Use logarithms to solve the equation. (Round answers to four decimal places.) x = __________

Graph the function.

How long will it take an investment to triple if it is continuously compounded at 15% per year? Select the correct answer rounded to the nearest year.

The given table corresponds to the function .

You are trying to determine the half-life of a new radioactive element you have isolated. You start with 3 grams, and 4 days later you determine that it has decayed down to 0.1 gram. What is the half-life? Round your answer to three decimal places. Select the answer rounded to three decimal places.

Graph the function.

How long, to the nearest year, will it take an investment in Japan to double its value if the interest is compounded every six months? Please round the answer to the nearest year. Country U.S. Japan Canada Korea Australia Yield 5.3% 1.5% 5.2% 5.4% 6.0%

Find the associated exponential decay model. when ; Half-life

Find the associated doubling time.

How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 20% per year? Round the answers to the nearest year. __________ years

The given table corresponds to the function .

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.

The amount of carbon-14 remaining in a sample that weighs A is given by where t is time in years. If tests on a fossilized skull reveal that 99.97% of the carbon-14 has decayed, how old is the skull? Select the correct answer rounded to the nearest integer.

Convert the exponential function to the form indicated. Round all coefficients to four significant digits.

Use logarithms to solve the equation. (Round answers to four decimal places.)