Math 127 Formula and Examples Sheet Exam #1 Average Rate Δ Y f(b) – f(a) of Change Δ X b – a Slope Intercept Form y-y0 = m(x-x0 ) Half Life (t 1/2 ) = - ln(2) Doubling Time = ln(2) k k Periodic Function = P0 a t a = e k k = ln(a) Continuous Function = P0 e kt r = a-1 a = r + 1 Decay when rate is negative, Growth rate is positive. EXAMPLES: 1. Prozac has a half-life of 3 days. What percent remains after a week? P(t) = P0 e-kt t 1/2 = ln(2) k = ln(2) k = -.23 k 3 P(7) = P0 e-k(7) = P0 (.198) ≈ P0 (.2), only 20% of original substance remains after a week. 2. A substance loses 10% of its mass in 5 hours. Find the half-life of this substance. (Exponential Decay) P(5) = P0 e-5k r = 1 - .1 = .9 (of original substance) .9P0 = P0 e-5k .9 = e-5k ln(.9) = ln(e-5k ) -5kln(e) = ln(.9) * NOTE: ln(e) = 1 k = ln(.9) = .021 t 1/2 = ln(2) = 32.9 hours 5 .021 3. A painting is supposedly by Vermeer (1632-1675) and it contains 99.5% of its carbon-14. Its half-life is 5,730 years. Is the painting fake? t
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This note was uploaded on 03/03/2010 for the course MATH 127 taught by Professor Rudvalis during the Fall '07 term at UMass (Amherst).