An unknown radioactive substance has a half-life of 124 years. How long will it take a 67 grams sample to degrade to 18 grams?

67(1/2)^(t/124) = 18

or
.5^(t/124) = .2686..
take log of both sides

(t/124)log .5 = log .2686..
t/124 = 1.896164...

t = appr 235 years

check:
after 124 years, there would be 33.5 g left
after 248 years , there would be 16.75 g
My answer is reasonable

To find out how long it will take for the radioactive substance to degrade from 67 grams to 18 grams, we can use the equation for exponential decay:

N(t) = N0 * (1/2)^(t / T),
where
N(t) is the current amount of the substance,
N0 is the initial amount of the substance,
t is the time that has passed, and
T is the half-life of the substance.

In this case, the initial amount N0 is 67 grams, and the final amount N(t) is 18 grams. We need to find the time t. Let's substitute the values into the equation:

18 = 67 * (1/2)^(t / 124).

Now, we can solve for t by isolating the variable:

(1/2)^(t / 124) = 18 / 67.

Taking the natural logarithm (ln) of both sides will help us simplify the equation:

ln[(1/2)^(t / 124)] = ln(18 / 67).

Using the logarithmic property, we can bring down the exponent:

(t / 124) * ln(1/2) = ln(18 / 67).

Now, isolate t by multiplying both sides by 124 and dividing by ln(1/2):

t = (124 * ln(18 / 67)) / ln(1/2).

Using a scientific calculator or a tool like a spreadsheet, we can calculate the value of t.

To calculate the time it takes for a radioactive substance to degrade, we need to use the half-life formula. The formula for radioactive decay is:

N = N0 * (1/2)^(t / half-life)

Where:
N is the final quantity of the substance
N0 is the initial quantity of the substance
t is the time passed
half-life is the time it takes for the substance to decay by half

In this case, we know that the initial quantity (N0) is 67 grams, and the final quantity (N) is 18 grams. The half-life is given as 124 years. We need to solve for 't', the time passed.

Let's plug in the values into the formula:

18 = 67 * (1/2)^(t / 124)

To isolate 't' and solve for it, we can use logarithms. Taking the logarithm of both sides of the equation is a common approach. In this case, we'll use the natural logarithm (ln):

ln(18) = ln(67 * (1/2)^(t / 124))

We can use the logarithmic rule that states ln(a * b) = ln(a) + ln(b), which enables us to separate the terms on the right side:

ln(18) = ln(67) + ln[(1/2)^(t / 124)]

Next, we can use another logarithmic rule that states ln(a^b) = b * ln(a):

ln(18) = ln(67) + (t / 124) * ln(1/2)

Now we can isolate 't' by moving the terms to one side:

(t / 124) * ln(1/2) = ln(18) - ln(67)

To solve for 't', we multiply both sides by 124 and divide by ln(1/2):

t = (124 * [ln(18) - ln(67)]) / ln(1/2)

Calculating this expression will give us the time it takes for the sample to degrade from 67 grams to 18 grams.