Question

A scientist measures the initial amount of Carbon-14 in a substance to be 25 grams. The relationship between A, the amount of Carbon-14 remaining in that substance, in grams, and t, the elapsed time, in years, since the initial measurement is modeled by the following equation: A = 25e^0.00012t. In how many years will the substance contain exactly 20 grams (g) of Carbon-14? Give an exact answer expressed as a natural logarithm. a) t = ln(20/25) / 0.00012 b) t = ln(25/20) / 0.00012 c) t = ln(20/25) * 0.00012 d) t = ln(25/20) * 0.00012

Answer 1

To find out in how many years the substance will contain exactly 20 grams of Carbon-14, you can use the given equation and solve for t:

A = 25e^(0.00012t)

You want to find t when A is 20 grams, so substitute A = 20 into the equation:

20 = 25e^(0.00012t)

Now, divide both sides by 25 to isolate the exponential term:

(20/25) = e^(0.00012t)

Now, take the natural logarithm (ln) of both sides to solve for t:

ln(20/25) = ln(e^(0.00012t))

Using the property of logarithms that ln(e^x) = x:

ln(20/25) = 0.00012t

Now, solve for t by dividing both sides by 0.00012:

t = ln(20/25) / 0.00012

So, the correct answer is:

a) t = ln(20/25) / 0.00012

Explanation: