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Transcript
8.3 Applications of Exponential Functions 3/25/2013 Compound Interest Interest that accrues on the initial principal and the accumulated interest of a principal deposit, loan or debt. Compounding of interest allows a principal amount to grow at a faster rate than simple interest, which is calculated as a percentage of only the principal amount. Compounding Interest Formula π ππ‘ π π‘ = ππ 1 + π Where P(t) = amount of money accumulated after t years, including interest. Po = principal amount (the initial amount you borrow or deposit) r = annual rate of interest (as a decimal) n = number of times the interest is compounded per year t = number of years the amount is deposited or borrowed for. An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4%, compounded quarterly. What is the balance after 6 years? ππ‘ π π π‘ = ππ 1 + π 4(6) .04 Po = $1,500 π π‘ = 1500 1 + 4 r = .04 24 =1500 1.01 n = 4 (quarterly = 4 times per year) = $1,904.60 t = 6 yrs Calculator: Follow order of Operations: Do whatβs in the ( ), then raise it to the exponent, then multiply by 1500. Exponential Growth Formula π‘ π π‘ = π0 (π) π Where P(t) = the amount of substance after time t Po = initial/starting amount b = growth factor = 2 for doubling = 3 for tripling t = time elapsed r = time it takes for growth to occur. Sarah observes that the number of bacteria in the colony in the lab doubles every 30mins. If the initial number of bacteria in the colony is 50, what is the total number of bacteria in the colony π‘ after 5 hours? π π‘ = π (π) π 0 5 Po = 50 b=2 t = 5 hrs r = .5 hrs (30mins) π π‘ = 50(2) .5 =50(2)10 =51,200 bacteria After 5 hrs, there are 51,200 bacteria Calculator: raise 2 to (5÷0.5) then multiply by 50. Half Life is the amount of time that the substance's total amount is halved. Exponential Decay Formula (half- life) π‘ π π‘ = π0 (π) π Where P(t) = the amount of substance left after time t Po = initial/starting amount d = decay factor = ½ for half-life t = time elapsed r = time it takes for decay to occur. Technitium-99m is a radioactive substance used to diagnose brain, thyroid liver and kidney diseases. This radioactive substance has a half life of 6 hours. If there are 200 mgs of this technetium-99m, how much will there be in 12 hours? π π‘ = π (π)π‘ π 0 12 Po = 200 mg d=½ t = 12 hrs r = 6 hrs π π‘ = 200(½) 6 =200(½)2 = 50mg After 12 hrs, thereβs 50mg left. Calculator: raise 0.5 to (12÷6 or 2) then multiply by 200. Homework: WS 8.3 do ALL