1. Modeling using Logarithms

2. Compound Interest
How long, to the nearest tenth of a year, will it take 12,500 dollars to grow to 20,000 dollars at a 6.5% annual interest compounded quarterly?

3. How long, to the nearest tenth of a year, will it take 50,000 dollars to triple in value at 7.5% annual interest rate compounded continuously?

4. What interest rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

5. Exponential Growth Model
A(t)= amount after time t
A0 = initial amount
K = rate of growth or decay (need other info sometimes to figure this out)
t= time

6. In 1970 the U. S population was 203. 3 million
In 1970 the U.S population was million. By 2010, it had grown to million. By what year will the U.S population reach 335 million.
Use the preliminary data from 1970 to 2010 to determine the growth rate (k). Then use the 1970 data to determine A(t) the population t years after 1970.

7. In 2000, the population of Africa was 807 million by 2011 it had grown to 1052 million. By which year will Africa’s population reach 2000 million? 2 billion?

8. Half Life
We use the same growth model, but now k will end up negative.
Carbon-14 decays exponentially with a half-life of approximately 5715 years. The half-life of a substance is the time required for half of a given sample to disintegrate. Thus, after 5715 years a given amount of carbon-14 will have decayed to half the original amount. Carbon dating is useful for artifacts or fossils up to 80,000 years old. Older objects do not have enough carbon-14 left to determine age accurately.

9. Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14.
Start with
Here we can substitute A0/2 for A(t).

10. Now divide both sides by Ao. This will then allow you to solve for k
Now divide both sides by Ao. This will then allow you to solve for k. So then substitute that back into the original growth model and thus you have the half-life model for carbon-14.
In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the scrolls.

11. Newton’s Law of Cooling
C is the constant temp of the surrounding medium
T0 is initial temp of heated object
K is a negative constant that is associated with the cooling object
A cake removed from the oven has a temp of 210 degrees. It is left to cool in a room that has temp of 70 degrees. After 30 minutes the temp of the cake is 140 degrees. Use Newton’s Law of Cooling to find a model for the temp. of the cake, T, after t minutes. When will the cake reach 90 degrees?