# Half-Life Problem: Solving A0e13K to Get K

• lwelch70
Divide both sides by 400:0.75 = e^(13k)Take the natural log of both sides:ln(0.75) = 13kDivide both sides by 13:k = ln(0.75)/13Now use this value to solve for t:300 = 400e^((ln(0.75)/13)t)Divide both sides by 400:0.75 = e^((ln(0.75)/13)t)Take the natural log of both sides:ln(0.75) = (ln(0.75)/13)tDivide both sides by ln(0.75)/13
lwelch70
1. Half life of a radioactive substance is 13 days. How long for 400 grams to decay to 300 grams? Solve algebraically and show all work. Give both exact answers and the answer rounded to 4 decimal places.

So I manipulated (1/2)A0=A0e13K to obtain K = ln(1/2)/13

Where do I go from here?

lwelch70 said:
1. Half life of a radioactive substance is 13 days. How long for 400 grams to decay to 300 grams? Solve algebraically and show all work. Give both exact answers and the answer rounded to 4 decimal places.

So I manipulated (1/2)A0=A0e13K to obtain K = ln(1/2)/13

Where do I go from here?

Put your known numbers back in the equation:

A = A0e(kt)

## 1. What is the Half-Life Problem?

The Half-Life Problem is a mathematical concept used in the field of radioactive decay. It refers to the amount of time it takes for half of a given amount of a radioactive substance to decay. It is an important concept in understanding the rate of decay and the stability of a substance.

## 2. How is the Half-Life Problem solved?

The Half-Life Problem is solved by using the equation A=A0e^-kt, where A is the amount of the substance remaining after time t, A0 is the initial amount, and k is the decay constant. By plugging in the values for A, A0, and t, the value for k can be solved for.

## 3. What is the significance of solving A0e^-kt to get k?

Solving A0e^-kt to get k is significant because it allows scientists to understand and predict the rate of decay for a radioactive substance. This information is important for various applications, such as in nuclear power plants or in radiocarbon dating.

## 4. How is the Half-Life Problem used in real-world scenarios?

The Half-Life Problem is used in various real-world scenarios, such as in medicine for determining the decay rate of radioactive isotopes used in treatments, in environmental science for understanding the half-life of pollutants, and in geology for dating the age of rocks and fossils.

## 5. Are there any limitations to using the Half-Life Problem?

One limitation of the Half-Life Problem is that it assumes a constant decay rate, which may not always be the case in real-world scenarios. Additionally, the Half-Life Problem does not take into account any external factors that may affect the rate of decay, such as temperature or pressure.

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