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# Interpreting time in exponential models

Sal finds the time interval over which a quantity changes by a given factor in various exponential models.

## Want to join the conversation?

• This section is explained really well and I'm happy to say I understand it well. My questions are:

1) What does the denominator of 't' in any of the modelled functions mentioned in the video actually mean? What is the purpose/reason/effect of 't' having a denominator ?

2) Throughout the video in any of the examples, 't' has always been made a value such that the power of 't' and it's denominator would be equal to 1, effectively "removing" the power. what effect would it have on the answer/function if the value of 't' was never a multiple of it's denominator?

This is my first time asking a question on Khan Academy, hope I have phrased it well enough. Will greatly appreciate a response
• It means that the common ratio were raised to a fraction of t.

Let say you have something that grows 50% every 3 days. Notice how it says every 3 days, and not every day. This means for every 3 days it increases by 50%. In term of d days, you would write 1.50^d, but this is not correct, because this means that every day it grows by 50%, so to make it correct, the d days has to be divided by 3, hence we write d/3, which makes it 1.50^(d/3)

Similarly, if say you have something that grows 50% every day. For d days with the equation 1.50^(d/24) this means it is calculated at every hour (24hrs in a day) not day or 1/24 of a day. So when you you get rid of the 24 to make a whole 1, you are turning it into 1 day and not hour anymore.

Hope this helps clarifying it.
• At , why does Sal say % after 0.125?
• He slipped up. He should've not said percent after 0.125 and it would've been fine.
• As a general rule, is it better to use decimals or fractions when doing problems like this? My personal preference is fractions, but could someone please point out the merits of using decimals? For example, if you look at , that table gets a bit messy. Is it better to use fractions in general or are there some cases where decimals are good?
• It's sometimes hard to immediately see the value of a fraction, where decimals show you very clearly the magnitude of the number, so decimals will be more useful in the real world in situations where you need to know what exactly a number is. I find fractions easier to deal with when manipulating expressions, though, and you always know you'll have the exact value when dealing with fractions where decimals sometimes have repeating numbers and so forth.

Basically, it really depends on what you're trying to do.
• At about minutes, Sal says that you would be multiplying by 1+4/5. Where did he get the 1 from?
• let "B" be number of branches at any point in time
then as the question suggest and I quote:" Howard's tree gains 4/5 branches every (______)years."
So its solution would go like this:
=>B + 4/5B
=>B( 1 + 4/5)
=> B(9/5)
and then you know the rest.
hope this helps!
• Uhm.. why do we have to divide " t " by some numbers? we didn't do that before...
• The reason its being divided is basically saying every 61.4 days 0.125% of the samples mass remains!(from the video)
hope this helped! :D
• What makes Math one of the most hardest subjects in school to learn about? What makes it super important?
• Wait, I don't completely understand that last question. When Sal explains that every 7.3 years the tree's branches increase by a ratio of 9/5. But 4/5 is a completely different ratio, and to my understanding should yield a different output..? At around 6 minutes and 25 seconds. (I got 5.84 when I evaluated this question myself, but I'd really like to know if I'm missing something :])
• Oops, I see my mistake. It's gaining 4/5 branches so when it's multiplied by 9/5 that is what's happening in the first place ,:D
• At , why did he have to put an exponent there? It was already there anyways, so did he just have to identify it to show what he was doing?
• He didn't need to, I believe he just wanted to make a point that t/5.5 had simplified to just 1.
• - Where did I go wrong with this problem? I tried to figure it out like this:

If the sample lost 87.5% from its initial 320 grams, then the amount left would be 320 - (320*0.875) = 40 grams after 87.5% decrease.

Now all that is left is to find what input of t provides an output of 40 for the function M(t)

M(t) = 320 * (0.125)^(t/61.4)
40 = 320 * (0.125)^(t/61.4)
I then used properties of exponents and properties of logarithms in order to solve for t and got t = 65... Did I set up the equation correctly or did I make a mistake when solving for t?

edit: Nevermind, I made a mistake when solving for t. Accidentally misused the property of exponents where a^(b-c) = (a^b)/(a^c). Got the right answer finally.