Exponential Decay And The Number

1) A radioactive substance decays at the rate of 3% per hour. This is an example of exponential decay. true or false

The given statement is true because the decay rate of 3% per hour indicates that the substance is undergoing exponential decay. In exponential decay, the quantity decreases by a fixed percentage over a fixed interval of time. In this case, the substance is decaying at a constant rate of 3% per hour, which is characteristic of exponential decay.

Explanation

The given statement is true because the decay rate of 3% per hour indicates that the substance is undergoing exponential decay. In exponential decay, the quantity decreases by a fixed percentage over a fixed interval of time. In this case, the substance is decaying at a constant rate of 3% per hour, which is characteristic of exponential decay.

Submit

2) Evaluate the expression: e³ ·e⁵

E^2

E^8

The expression e3 ·e5 can be simplified by multiplying the exponents. Since the base is the same (e), we can add the exponents together to get e^8. Therefore, the correct answer is e^8.

Explanation

The expression e3 ·e5 can be simplified by multiplying the exponents. Since the base is the same (e), we can add the exponents together to get e^8. Therefore, the correct answer is e^8.

3) 3^0 = ?

3

0

1

The expression 3^0 represents the power of 3 raised to the exponent of 0. Any number raised to the power of 0 is equal to 1. Therefore, 3^0 is equal to 1.

Explanation

The expression 3^0 represents the power of 3 raised to the exponent of 0. Any number raised to the power of 0 is equal to 1. Therefore, 3^0 is equal to 1.

4) An exponential decay function becomes smaller and smaller until it reaches 0. true or false

An exponential decay function does not necessarily become smaller and smaller until it reaches 0. It approaches but never actually reaches 0. As the input values increase, the output values become smaller and smaller, but they never reach 0. Therefore, the statement is false.

Explanation

An exponential decay function does not necessarily become smaller and smaller until it reaches 0. It approaches but never actually reaches 0. As the input values increase, the output values become smaller and smaller, but they never reach 0. Therefore, the statement is false.

Submit

5) If e is to the power of -x then the function is a decay function. true or false

True

False

If the function is represented by e raised to the power of -x, then it is a decay function. This is because as the value of x increases, the exponent becomes more negative, causing the function to approach zero. In other words, the function decreases or decays as x increases. Therefore, the given answer "true" is correct.

Explanation

If the function is represented by e raised to the power of -x, then it is a decay function. This is because as the value of x increases, the exponent becomes more negative, causing the function to approach zero. In other words, the function decreases or decays as x increases. Therefore, the given answer "true" is correct.

6) Which of the following is an exponential decay function?

Y=3^x

Y=3^(-x)

An exponential decay function is a function in which the base is greater than 1 and the exponent is negative. In the given options, y=3^(-x) fits this criteria as it has a base of 3 (greater than 1) and the exponent is negative (-x). This means that as x increases, the value of y decreases exponentially. On the other hand, y=3^x represents exponential growth as the exponent is positive. Therefore, y=3^(-x) is the correct answer for an exponential decay function.

Explanation

An exponential decay function is a function in which the base is greater than 1 and the exponent is negative. In the given options, y=3^(-x) fits this criteria as it has a base of 3 (greater than 1) and the exponent is negative (-x). This means that as x increases, the value of y decreases exponentially. On the other hand, y=3^x represents exponential growth as the exponent is positive. Therefore, y=3^(-x) is the correct answer for an exponential decay function.

7) Nadia put $500 in her bank account with an interest compounded annually at the rate of 10%. This is an example of exponential growth. true or false

When money is deposited in a bank account with an interest rate compounded annually, the amount of money in the account grows exponentially over time. In this case, Nadia deposited $500 with an interest rate of 10%, meaning that her money will increase by 10% each year. This increase is compounded annually, resulting in exponential growth of her account balance. Therefore, the statement that this is an example of exponential growth is true.

Explanation

When money is deposited in a bank account with an interest rate compounded annually, the amount of money in the account grows exponentially over time. In this case, Nadia deposited $500 with an interest rate of 10%, meaning that her money will increase by 10% each year. This increase is compounded annually, resulting in exponential growth of her account balance. Therefore, the statement that this is an example of exponential growth is true.

Submit

8) Nadia put $500 in her bank account with an interest compounded annually at the rate of 10%.
The model for this problem is
a)y = 500 (1.010)^x

OR

b) y = 500 (1.10)^x

The correct answer is b,b),B. The model for this problem is y = 500 (1.10)x. This model represents the amount of money in Nadia's bank account after x years with an interest rate of 10% compounded annually. The initial amount of money is $500 (y = 500) and it grows exponentially over time with the factor (1.10) representing the 10% interest rate. Therefore, the correct answer is b,b),B which corresponds to the correct model equation.

Explanation

The correct answer is b,b),B. The model for this problem is y = 500 (1.10)x. This model represents the amount of money in Nadia's bank account after x years with an interest rate of 10% compounded annually. The initial amount of money is $500 (y = 500) and it grows exponentially over time with the factor (1.10) representing the 10% interest rate. Therefore, the correct answer is b,b),B which corresponds to the correct model equation.

Submit

9) If you take 400 mg of ibuprofen and every hour 29% of ibuprofen is broken down in the body, how much ibuprofen is in your body system after 3 hours? Round to whole number (without decimals) without units.

not-available-via-ai

Explanation

not-available-via-ai

Submit

10) An exponential decay function always has -x as its power. true or false

An exponential decay function does not always have -x as its power. The power in an exponential decay function is typically a negative value, but it can be any negative constant, not just -x. Therefore, the statement is false.

Explanation

An exponential decay function does not always have -x as its power. The power in an exponential decay function is typically a negative value, but it can be any negative constant, not just -x. Therefore, the statement is false.

Submit

11) The length l (in cm) of a tiger shark is modeled by the function
l = 337 - 276e^-0.178t
where t is the shark’s age in years.

which of the following is true?

The shark is 201 cm long when it is 5 years old.

The shark is 223 cm long when it is 10 years old.

The shark is 144 cm long when it is 2 years old.

hint: to check if A is the correct answer

put 5 instead of t in the equation and check if the length is 201

Explanation

hint: to check if A is the correct answer

put 5 instead of t in the equation and check if the length is 201

12) A radioactive substance decays at the rate of 3% per hour. If we have 300 g of this substance which of the following would be true?

Only 213 g would remain after 3 hours

Only 252 g would remain after 2 hours

Only 257 g would remain after 5 hours

The correct answer is "only 257 g would remain after 5 hours." This can be determined by calculating the decay of the substance over time. Since the substance decays at a rate of 3% per hour, after 1 hour, 97% of the substance would remain (300 g * 0.97 = 291 g). After 2 hours, 94% of the substance would remain (291 g * 0.97 = 282.27 g). After 3 hours, 91% of the substance would remain (282.27 g * 0.97 = 273.70 g). After 4 hours, 88% of the substance would remain (273.70 g * 0.97 = 265.89 g). Finally, after 5 hours, 85% of the substance would remain (265.89 g * 0.97 = 257.69 g), which rounds down to 257 g.

Explanation

The correct answer is "only 257 g would remain after 5 hours." This can be determined by calculating the decay of the substance over time. Since the substance decays at a rate of 3% per hour, after 1 hour, 97% of the substance would remain (300 g * 0.97 = 291 g). After 2 hours, 94% of the substance would remain (291 g * 0.97 = 282.27 g). After 3 hours, 91% of the substance would remain (282.27 g * 0.97 = 273.70 g). After 4 hours, 88% of the substance would remain (273.70 g * 0.97 = 265.89 g). Finally, after 5 hours, 85% of the substance would remain (265.89 g * 0.97 = 257.69 g), which rounds down to 257 g.

13)

choose the equation that best describes the graph

Y= - (0.25)^x

Y=(0.25)^x

Y = (0.5)^x

The correct answer is y=(0.25)^x. This equation represents an exponential decay function, where the base is less than 1. As x increases, the value of (0.25)^x decreases rapidly. This is reflected in the graph, where the y-values decrease exponentially as x increases. The other two equations, y=-(0.25)^x and y=(0.5)^x, do not accurately describe the graph as they either have a negative sign or a base greater than 1, which would result in exponential growth or different patterns.

Explanation

The correct answer is y=(0.25)^x. This equation represents an exponential decay function, where the base is less than 1. As x increases, the value of (0.25)^x decreases rapidly. This is reflected in the graph, where the y-values decrease exponentially as x increases. The other two equations, y=-(0.25)^x and y=(0.5)^x, do not accurately describe the graph as they either have a negative sign or a base greater than 1, which would result in exponential growth or different patterns.

14) Which of the following is an exponential function?

Y=3^2

Y=3x

Y=(1/3)^x

An exponential function is a mathematical function in which the independent variable is in the exponent. In the given options, only the equation y=(1/3)^x fits this definition. The variable x is in the exponent, indicating that the function is exponential. The base of the function is 1/3, which means that as x increases, the value of y will decrease exponentially.

Explanation

An exponential function is a mathematical function in which the independent variable is in the exponent. In the given options, only the equation y=(1/3)^x fits this definition. The variable x is in the exponent, indicating that the function is exponential. The base of the function is 1/3, which means that as x increases, the value of y will decrease exponentially.

15) A radioactive substance decays at the rate of 3% per hour. If we have 200 g of this substance and after some time only about 147 g of it remains, how much time has passed?

8 hours

10 hours

12 hours

If a radioactive substance decays at a rate of 3% per hour, it means that after each hour, 97% of the substance remains. To find out how much time has passed, we can set up an equation:

200g * (0.97)^t = 147g

where t represents the number of hours that have passed. Solving this equation, we find that t is approximately 10 hours. Therefore, the correct answer is 10 hours.

Explanation

If a radioactive substance decays at a rate of 3% per hour, it means that after each hour, 97% of the substance remains. To find out how much time has passed, we can set up an equation:

200g * (0.97)^t = 147g

where t represents the number of hours that have passed. Solving this equation, we find that t is approximately 10 hours. Therefore, the correct answer is 10 hours.

16)

Which of the following is true?

The answer to 18 is e^-2

The answer to 21 is 3e^4

The answer to 23 is e^(2x+5)

The correct answer states that the answer to 21 is 3e^4. This means that when evaluating an expression or equation where the variable is equal to 21, the result will be 3 multiplied by e raised to the power of 4. The value of e is a mathematical constant approximately equal to 2.71828. Therefore, the answer to 21 is 3 multiplied by 2.71828 raised to the power of 4.

Explanation

The correct answer states that the answer to 21 is 3e^4. This means that when evaluating an expression or equation where the variable is equal to 21, the result will be 3 multiplied by e raised to the power of 4. The value of e is a mathematical constant approximately equal to 2.71828. Therefore, the answer to 21 is 3 multiplied by 2.71828 raised to the power of 4.

17) The length l (in cm) of a tiger shark is modeled by the function
l = 337 - 276e^-0.178t
where t is the shark’s age in years.

what is the length of the shark when it is 7 years old?

256.6 cm

257.61 cm

258 cm

The length of the shark when it is 7 years old can be found by substituting t = 7 into the given function. By plugging in the value, we get l = 337 - 276e^(-0.178*7). Evaluating this expression, we find that the length is approximately 257.61 cm. Therefore, the correct answer is 257.61 cm.

Explanation

The length of the shark when it is 7 years old can be found by substituting t = 7 into the given function. By plugging in the value, we get l = 337 - 276e^(-0.178*7). Evaluating this expression, we find that the length is approximately 257.61 cm. Therefore, the correct answer is 257.61 cm.

18)

Which of the following is true?

The answer to 17 is e^2

The answer to 19 is e^(3x) / 3

The answer to 20 is e^8x / 9

The correct answer is "the answer to 19 is e^(3x) / 3". This means that when solving the equation or problem labeled as "19", the result should be e raised to the power of 3x, divided by 3.

Explanation

The correct answer is "the answer to 19 is e^(3x) / 3". This means that when solving the equation or problem labeled as "19", the result should be e raised to the power of 3x, divided by 3.

19) Make a table for the equation y = (2/3)^x -1

x->0,1,2
y->?,?,?

Y->1,2,3

Y->0,1,2

Y->0,-.33,-.56

The given table represents the values of y for different values of x in the equation y = (2/3)^x - 1. By substituting the given values of x (0, 1, 2) into the equation, we can calculate the corresponding values of y. The answer provided, y->0, -.33, -.56, represents the calculated values of y for x = 0, 1, 2 respectively.

Explanation

The given table represents the values of y for different values of x in the equation y = (2/3)^x - 1. By substituting the given values of x (0, 1, 2) into the equation, we can calculate the corresponding values of y. The answer provided, y->0, -.33, -.56, represents the calculated values of y for x = 0, 1, 2 respectively.

20)

Which of the following is true?

#49, #51, #52 are all decay functions

#51, #53, #54 are all growth functions

#54, #55, #56 are all decay functions

None of the above are true

not-available-via-ai

Explanation

not-available-via-ai

21)

Which of the following is true?

The answer to #33 is 20.1

The answer to #35 is 5.47

The answer to #37 is .25

The answer to #39 is 2981

The answer to #34 is .513

The given answer states that the answer to question #34 is .513.

Explanation

The given answer states that the answer to question #34 is .513.

22) The natural base e is a rational number. choose the best answer from below.

True, it is a decimal number so its rational.

False, it is a decimal number and a fraction

True, it cannot be divided by any number

False, it is a decimal that cannot be written as a fraction

The given correct answer is false. This is because the natural base e, which is approximately equal to 2.71828, is not a rational number. A rational number can be expressed as a fraction, but e cannot be written as a fraction. Therefore, it is a decimal number and not a fraction.

Explanation

The given correct answer is false. This is because the natural base e, which is approximately equal to 2.71828, is not a rational number. A rational number can be expressed as a fraction, but e cannot be written as a fraction. Therefore, it is a decimal number and not a fraction.

Submit