Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?
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Transcript of Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?
![Page 1: Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?](https://reader036.fdocuments.in/reader036/viewer/2022071808/5681322c550346895d988f18/html5/thumbnails/1.jpg)
Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?
![Page 2: Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?](https://reader036.fdocuments.in/reader036/viewer/2022071808/5681322c550346895d988f18/html5/thumbnails/2.jpg)
Better than investing annually?
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e (2.71828…): the natural base
It represents the base rate of growth shared by all continuously
growing processes.
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e (2.71828…): the natural base
Q: Why the natural base?A: Because it shows up in
population growth, radioactive decay, and in systems that exhibit
continuous growth or decay.
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Suppose we applied a (theoretical) continuous growth rate to an
investment.
Continuously Compounding
Interest Formula:
A = Pert
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A = Pert
Note that the expression has
been substituted with
er now.1
lim 1n
ne
n
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Take Tacoma’s population (as of 2012) of 202,010 which was growing at
1.8%. If the growth rate remains the same, what will its population be in
2013?Continuously
Compounding Interest Formula:A = Pert
= 202,010e0.018*3
= ~213218
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It’s e-asy!
If Portland, with its 2012 population of 603,106, is continuously growing at
1.7% then what will its population be in 2017?
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How e-nterestingPortland, with its 2012 population of 603,106, is continuously growing at 1.7%. When what will its population surpass 700,000?
700,000 = 603,106e0.017t [solving for t!]
1.16065… = e0.017t [divide by 603,106]
loge1.16065 = 0.017t [put into log form, and with a base of e]
0.148987… = 0.017t [evaluate dat log]
8.76 396… = t [divide by 0.017]
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e-nsane in the Membrane
Portland, with its 2012 population of 603,106, is continuously growing at 1.7%.
When what will its population surpass 800,000?
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oh btw
logex = ln(x)
The natural basedeserves a natural logarithm.
So from now on, instead of using loge(3),a shortcut to use is now ln(3).