To solve this problem, we can use the formula for exponential decay, which is:

N = N0 * e^(rt)

where:
N is the final amount of the substance,
N0 is the initial amount of the substance,
r is the decay rate, and
t is the time.

In this case, we know that N = 3 grams, N0 = 15 grams, and r = -0.03 (since it's a decay rate of 3% per year). We want to find t.

So we can set up the equation as follows:

3 = 15 * e^(-0.03t)

To solve for t, we first divide both sides by 15:

3/15 = e^(-0.03t)

0.2 = e^(-0.03t)

Then we take the natural logarithm (ln) of both sides:

ln(0.2) = ln(e^(-0.03t))

ln(0.2) = -0.03t

Finally, we solve for t by dividing both sides by -0.03:

t = ln(0.2) / -0.03

Using a calculator, we find that t ≈ 61.1 years.

So it will take approximately 61.1 years for the element to decay from 15 grams to 3 grams.

Question

To solve this problem, we can use the formula for exponential decay, which is:

N = N0 * e^(rt)

where:
N is the final amount of the substance,
N0 is the initial amount of the substance,
r is the decay rate, and
t is the time.

In this case, we know that N = 3 grams, N0 = 15 grams, and r = -0.03 (since it's a decay rate of 3% per year). We want to find t.

So we can set up the equation as follows:

3 = 15 * e^(-0.03t)

To solve for t, we first divide both sides by 15:

3/15 = e^(-0.03t)

0.2 = e^(-0.03t)

Then we take the natural logarithm (ln) of both sides:

ln(0.2) = ln(e^(-0.03t))

ln(0.2) = -0.03t

Finally, we solve for t by dividing both sides by -0.03:

t = ln(0.2) / -0.03

Using a calculator, we find that t ≈ 61.1 years.

So it will take approximately 61.1 years for the element to decay from 15 grams to 3 grams.

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