504 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions EXAMPLE 4 Determining an Exponential Function to Model Radioactive Decay Suppose 600 g of a radioactive substance are present initially and 3 yr later only 300 g remain. (a) Determine an exponential function that models this decay. (b) How much of the substance will be present after 6 yr? SOLUTION (a) We use the given values to find k in the exponential equation y = y0 ekt. Because the initial amount is 600 g, y0 = 600, which gives y = 600ekt. The initial amount (600 g) decays to half that amount (300 g) in 3 yr, so its halflife is 3 yr. Now we solve this exponential equation for k. y = 600ekt Let y 0 = 600. 300 = 600e3k Let y = 300 and t = 3. 0.5 = e3k Divide by 600. ln 0.5 = ln e3k Take the natural logarithm on each side. ln 0.5 = 3k ln e x = x, for all x. ln 0.5 3 = k Divide by 3. k ≈ -0.231 Use a calculator. A function that models the situation is y = 600e-0.231t. (b) To find the amount present after 6 yr, let t = 6. y = 600e-0.231t Model from part (a) y = 600e-0.231162 Let t = 6. y = 600e-1.386 Multiply. y ≈150 Use a calculator. After 6 yr, 150 g of the substance will remain. S Now Try Exercise 19. EXAMPLE 5 Solving a Carbon Dating Problem Carbon-14, also known as radiocarbon, is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists can determine the age of the remains by comparing the amount of radiocarbon with the amount present in living plants and animals. This technique is called carbon dating. The amount of radiocarbon present after t years is given by y = y0 e-0.0001216t, where y0 is the amount present in living plants and animals. (a) Find the half-life of carbon-14. (b) Charcoal from an ancient fire pit on Java contained 1 4 the carbon-14 of a living sample of the same size. Estimate the age of the charcoal.
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