1. 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

2. 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
= 550 𝑒 𝑟
1.82= 𝑒 5𝑟
ln =𝑙𝑛 𝑒 5𝑟
ln =5𝑟
ln = 5𝑟 5
𝑟=0.12
12% interest rate

3. 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.

4. 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

5. 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

6. 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 −24,110 =𝑘 𝑘≈
Pu-239 is decaying at a constant of atoms per second.

7. 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 − 𝑡
Now we solve for t to answer the question.
𝑁 𝑡 =0.1
𝑁 0 =1 𝑘=

8. 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 − 𝑡
𝑙𝑛0.1=𝑙𝑛 𝑒 − 𝑡
𝑙𝑛0.1=− 𝑡
𝑙𝑛0.1 − =𝑡
𝑡≈80,000
It takes approximately 80,000 years for 1 gram of Pu-239 to decay to 0.1 grams.

9. 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
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

10. 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 grams of cesium left after 90 years.

11. 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

12. 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 =−0.0042𝑡
ln − =𝑡
𝑡= 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
0.25=8 𝑒 − 𝑡
= 𝑒 − 𝑡
= 𝑒 −0.0042𝑡
ln =𝑙𝑛 𝑒 −0.0042𝑡