31. Radioactive decay of material is determined using the formula N = Noe^-kt. Find the constant k for Plutonium-239 given that its half-life is 24,390 years.

==> I don't really get where to start here because the problem does not give you how much Plutonium-239 there is. I just need a hint, no need to work out the entire problem for me please :)

35. The exponential growth rate of the population of Europe west of Russia is 1% per year. What is the doubling time?

==> I just used the rule of 72 to estimate it out to be about 72 years, but I don't think this is the correct process. Can anyone help me on these? Thanks so much!! :)

For both questions, you can solve for the constant "k" by using the given half-life or growth rate. Here's how you can approach each question:

31. Radioactive decay of material is determined using the formula N = Noe^-kt. In this case, we want to find the constant "k" for Plutonium-239 with a half-life of 24,390 years.

To find "k," we need to use the half-life information. The half-life is the time it takes for half of the original quantity to decay.

- Start with the formula: N = Noe^-kt
- Let's say "No" is the original amount, and "N" is the final amount after the given time.
- Since the half-life is the time it takes for N to become half of No, we can write the following equation: N = 0.5No

Now, substitute these values into the formula and solve for "k":

0.5No = Noe^(-kt)

Divide both sides by No:

0.5 = e^(-kt)

To isolate "k," take the natural logarithm of both sides:

ln(0.5) = ln(e^(-kt))

Apply the logarithmic property ln(e^x) = x:

ln(0.5) = -kt

Finally, solve for "k":

k = -ln(0.5) / t

Plug in the given half-life of Plutonium-239 (24,390 years) to find the value of "k."

35. The exponential growth rate of the population of Europe west of Russia is 1% per year. We want to find the doubling time.

To find the doubling time, you need to determine how long it takes for the population to double at the given growth rate of 1% per year.

Using the exponential growth formula A = P(1 + r)^t, where:
- A is the final amount
- P is the initial amount
- r is the growth rate (as a decimal)
- t is the time

In this case, we know that the growth rate, r, is 1% or 0.01. We want to find the time it takes for the population to double, so A = 2P.

Plug in these values into the formula and solve for the time, t:

2P = P(1 + 0.01)^t

Simplify the equation:

2 = (1.01)^t

Take the logarithm of both sides to isolate "t":

log(2) = log((1.01)^t)

Apply the logarithmic property log(a^b) = b*log(a):

log(2) = t * log(1.01)

Finally, solve for "t":

t = log(2) / log(1.01)

Use logarithmic values to calculate the doubling time.

Remember that logarithms can be calculated using scientific calculators or software tools with logarithmic functions.