Warm up Honors algebra 2 2/25/19 1. Your 3 year investment of $20,000 received 5.2% interest compounded semi annually. What is your total return? 2. You borrowed $59,000 for 2 years at 11% interest compounded continuously. What total will you pay back? Go over homework

Suppose that the present value of $1000 to be received in 5 years is $550. What rate of interest, compounded continuously, was used to compute this present value? 𝐴=𝑃 𝑒 𝑟𝑡 1000=550 𝑒 𝑟 5 1000 550 = 550 𝑒 𝑟 5 550 1.82= 𝑒 5𝑟 ln 1.82 =𝑙𝑛 𝑒 5𝑟 ln 1.82 =5𝑟 ln 1.82 5 = 5𝑟 5 𝑟=0.12 12% interest rate

Half life Half life of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of decay. Natural decay is modeled by the function: 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 𝑁 𝑡 is the amount remaining k is the decay constant 𝑁 0 is the initial amount (at 𝑡=0) t is the time Decay constant is the fraction of the number of atoms that decay in 1 second.

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 grams? How would you solve this problem? Find your formula: 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 Find the decay constant (k) since it is not given. Write the decay function (with k) and solve for t

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 grams? 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 1 2 = 1 𝑒 −𝑘 24110 Find the decay constant for Pu-239. 𝑁 𝑡 =1/2 because ½ of the substance is remaining the same and the other half is decaying or changing. 𝑁 0 =1 𝑡=24,110

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 grams? 1 2 = 𝑒 −𝑘 24110 𝑙𝑛 1 2 =𝑙𝑛 𝑒 −𝑘 24110 𝑙𝑛 1 2 =−24,110𝑘 ln 1 2 −24,110 =𝑘 𝑘≈0.000029 Pu-239 is decaying at a constant of 0.000029 atoms per second.

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 grams? 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 0.1=1 𝑒 −0.000029𝑡 Now we solve for t to answer the question. 𝑁 𝑡 =0.1 𝑁 0 =1 𝑘=0.000029

Plutonium-239 (Pu-239) has a half-life of 24,110 years Plutonium-239 (Pu-239) has a half-life of 24,110 years. How long does it take for a 1 gram sample of Pu-239 to decay to 0.1 grams? 0.1= 𝑒 −0.000029𝑡 𝑙𝑛0.1=𝑙𝑛 𝑒 −0.000029𝑡 𝑙𝑛0.1=−0.000029𝑡 𝑙𝑛0.1 −0.000029 =𝑡 𝑡≈80,000 It takes approximately 80,000 years for 1 gram of Pu-239 to decay to 0.1 grams.

An isotope of cesium has a half life of 30 years. If 1 An isotope of cesium has a half life of 30 years. If 1.0 grams of cesium disintegrates over a period of 90 years, how many grams of cesium would remain? 𝑙𝑛 1 2 −30 = −30𝑘 −30 𝑘=0.0231 0.0231 is the decay constant for cesium 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 1 2 =1 𝑒 −𝑘(30) 𝑙𝑛 1 2 =𝑙𝑛 𝑒 −𝑘(30) 𝑙𝑛 1 2 =−30𝑘

An isotope of cesium has a half life of 30 years. If 1 An isotope of cesium has a half life of 30 years. If 1.0 grams of cesium disintegrates over a period of 90 years, how many grams of cesium would remain? 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 𝑁 𝑡 =1 𝑒 −0.0231(90) 𝑁 𝑡 =0.125 There will be 0.125 grams of cesium left after 90 years.

Polonium-214 has a relatively short half-life of 164 seconds Polonium-214 has a relatively short half-life of 164 seconds. How many seconds would it take for 8.0 g of this isotope to decay to 0.25 g? 𝑁 𝑡 = 𝑁 0 𝑒 −𝑘𝑡 1 2 = 𝑒 −𝑘 164 𝑙𝑛 1 2 = 𝑙𝑛𝑒 −𝑘 164 𝑙𝑛 1 2 =−164𝑘 𝑙𝑛 1 2 −164 =𝑘 𝑘=0.0042

Polonium-214 has a relatively short half-life of 164 seconds Polonium-214 has a relatively short half-life of 164 seconds. How many seconds would it take for 8.0 g of this isotope to decay to 0.25 g? ln 0.03125 =−0.0042𝑡 ln 0.03125 −0.0042 =𝑡 𝑡=825.18 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 0.25=8 𝑒 −0.0042 𝑡 0.25 8 = 𝑒 −0.0042 𝑡 0.03125= 𝑒 −0.0042𝑡 ln 0.03125 =𝑙𝑛 𝑒 −0.0042𝑡