Convert the given exponential function to the form indicated. Round all coefficients...

Convert the given exponential function to the form indicated. Round all coefficients to four significant digits. HINT [See Example 4 “Before we go on.”]

Example 4:

a. The weight of carbon 14 that remains in a sample that originally contained A grams is given by

where t is time in years. Find the half-life, the time it takes half of the carbon 14 in a sample to decay.

b. Repeat part (a) using the following alternative form of the exponential model in part (a):

c. Another radioactive material has a half-life of 7,000 years. Find an exponential decay model in the form

for the amount of undecayed material remaining. (The constant k is called the decay constant.)

d. How long will it take for 99.95% of the substance in a sample of the material in part (c) to decay?

Solution

a. We want to find the value of t for which C(t) = the weight of undecayed carbon 14 left _ half the original weight = 0.5A. Substituting, we get

Dividing both sides by A gives

b. This is similar to part (a): We want to solve the equation

for t. Dividing both sides by A gives

Taking the natural logarithm of both sides gives

as we obtained in part (a).

c. This time we are given the half-life, which we can use to find the exponential model the amount of radioactive material is

Because half of the sample decays in 7,000 years, this sample will decay to 0.5A grams in 7,000 years (t = 7,000). Substituting this information gives

Canceling A and taking natural logarithms (again using Identity 3) gives

ln(0.5) = −7,000k

so the decay constant k is

Therefore, the model is

d. If 99.95% of the substance in a sample has decayed, then the amount of undecayed material left is 0.05% of the original amount, or 0.0005A. We have