A 10 kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.

Solve this for k:

3 = 10 e^(-kt)
where t = 17 years

ln (3/10) = -17k

k = 0.0798 y^-1

Well, this is quite a radioactive question! To find the decay constant of the isotope, we can use the formula:

N(t) = N(0) * e^(-λt)

Where N(t) is the amount of the isotope at time t, N(0) is the initial amount, e is Euler's number (approximately 2.71828), λ is the decay constant, and t is the time.

In this case, we know that the initial amount N(0) is 10 kg, and the amount after 17 years N(t) is 3 kg. Let's plug these values into the equation:

3 = 10 * e^(-17λ)

To solve for λ, we need to do a little math here. Divide both sides by 10:

0.3 = e^(-17λ)

Now, let's take the natural logarithm (ln) of both sides:

ln(0.3) = -17λ

Finally, divide both sides by -17 to solve for λ:

λ = ln(0.3) / -17

And there you have it! The decay constant of the radioactive isotope is approximately equal to ln(0.3) divided by -17. Just don't let it go to your head, or you may start to feel a bit "radioactive" yourself!

To find the decay constant of the isotope, we can use the formula for exponential decay:

N(t) = N₀ * e^(-λt)

Where:
N(t) = the final quantity of the isotope (3 kg)
N₀ = the initial quantity of the isotope (10 kg)
λ = the decay constant
t = the time (17 years)

Now, we can rearrange the formula to solve for λ:

e^(-λt) = N(t)/N₀

Substituting the given values:

e^(-λ*17) = 3/10

Next, take the natural logarithm (ln) of both sides to eliminate the exponential function:

ln(e^(-λ*17)) = ln(3/10)

Simplifying the left side:

-17λ = ln(3/10)

Finally, divide both sides by -17 to solve for λ:

λ = ln(3/10) / -17

Using a calculator to evaluate this expression, the value of the decay constant (λ) is approximately -0.0622 per year.

To find the decay constant of the radioactive isotope, we can use the formula for exponential decay:

N(t) = N0 * e^(-λt)

where:
N(t) is the amount of the isotope at time t
N0 is the initial amount of the isotope (10 kg in this case)
e is the mathematical constant approximately equal to 2.71828
λ is the decay constant (what we want to find)
t is the time elapsed (17 years in this case)

Since we know that the isotope decays from 10 kg to 3 kg after 17 years, we can set up the following equation:

3 kg = 10 kg * e^(-λ * 17 years)

Now, we need to isolate the decay constant (λ). Divide both sides of the equation by 10 kg:

3 kg / 10 kg = e^(-λ * 17 years)

0.3 = e^(-λ * 17 years)

To isolate λ, we can take the natural logarithm of both sides of the equation:

ln(0.3) = ln(e^(-λ * 17 years))

The natural logarithm of e^(-λ * 17 years) simplifies to (-λ * 17 years) by the properties of logarithms.

ln(0.3) = -λ * 17 years

Now, divide both sides of the equation by (-17 years):

ln(0.3) / (-17 years) = λ

Using a calculator, we can evaluate the left side of the equation:

ln(0.3) / (-17 years) ≈ 0.1015 years^(-1)

Therefore, the decay constant of the isotope is approximately 0.1015 years^(-1).