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^(-0.00012 t) In how many years will the substance contain exactly 20 grams (g) of Carbon-14? Give an exact answer expressed as a natural logarithm. years

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^(-0.00012 t) In how many years will the substance contain exactly 20 grams  (g) of Carbon-14? Give an exact answer expressed as a natural logarithm. years

Bytelearn · Accepted Answer

Set up equation: Set up the equation with the given amount of Carbon-14. We are given the equation A = 25e^(-0.00012t) and we want to find the value of t when A = 20 grams. So, we set up the equation: 20 = 25e^(-0.00012t).Divide and isolate: Divide both sides of the equation by 25 to isolate the exponential term. 20 / 25 = 25e^(-0.00012t) / 25 0.8 = e^(-0.00012t)Take natural logarithm: Take the natural logarithm of both sides to solve for t. ln(0.8) = ln(e^(-0.00012t)) Using the property of logarithms that ln(e^x) = x, we get: ln(0.8) = -0.00012tSolve for t: Divide both sides by -0.00012 to solve for t. t = ln(0.8) / -0.00012Calculate value of t: Calculate the value of t using the natural logarithm. t = ln(0.8) / -0.00012 This is the exact answer expressed as a natural logarithm.

Full solution

More problems from Convert between exponential and logarithmic form: all bases

Related topics