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  • - [Instructor] We are told a phone sells for $600

  • and loses 25% of its value per year.

  • Write a function that gives the phone's value, V of t,

  • so value as a function of time, t years after it is sold.

  • So pause this video, and have a go at that

  • before we work through it together.

  • All right, so let's just think about it a little bit.

  • And I could even set up a table

  • to think about what is going on.

  • So this is t, and this is the value of our phone

  • as a function of t.

  • So it sells for $600.

  • At time t equals zero, what is V of zero?

  • Well it's going to be equal to $600.

  • That's what it sells for at time t equals zero.

  • Now at t equals one, what's going to happen?

  • Well it says that the phone loses 25%

  • of its value per year.

  • Another way to rewrite that it loses 25% of its value

  • per year is that it retains, it retains,

  • 100% minus 25% of its value per year,

  • or it retains 75%

  • of its value per year, per year.

  • So how much is it going to be worth after one year?

  • Well it's going to be worth $600,

  • $600 times 75%.

  • Now what about year two?

  • Well it's going to be worth what it was in year one

  • times 75% again.

  • So it's going to be $600 times 75% times 75%.

  • And so you could write that as times 75% squared.

  • And I think you see a pattern.

  • In general, if we have gone, let's just call it t years,

  • well then the value of our phone,

  • if we're saying it in dollars, is just going to be $600

  • times, and I could write it as a decimal,

  • 0.75, instead of 75%, to the t power.

  • So V of t is going to be equal to 600

  • times 0.75 to the t power, and we're done.

  • Let's do another example.

  • So here, we are told that a biologist has a sample

  • of 6,000 cells.

  • The biologist introduces a virus that kills 1/3

  • of the cells every week.

  • Write a function that gives the number of cells remaining,

  • which would be C of t, the cells as a function of time,

  • in the sample t weeks after the virus is introduced.

  • So again, pause this video

  • and see if you can figure that out.

  • All right, so I'll set up another table again.

  • So this is time, it's in weeks,

  • and this is the number of cells, C.

  • We could say it's a function of time.

  • So time t equals zero, when zero weeks have gone by,

  • we have 6,000 cells.

  • That's pretty clear.

  • Now after one week, how many cells do we have?

  • What's C of one?

  • Well it says that the virus kills 1/3

  • of the cells every week, which is another way of saying

  • that 2/3 of the cells are able to live

  • for the next week.

  • And so after one week, we're going to have 6,000 times 2/3.

  • And then after two weeks, or another week goes by,

  • we're gonna have 2/3 of the number that we had

  • after one week.

  • So we're gonna have 6,000 times 2/3 times 2/3,

  • or we could just write that as 2/3 squared.

  • So once again, you are likely seeing the pattern here.

  • We are going to, at time t equals zero, we have 6,000,

  • and then we're going to multiply by 2/3

  • however many weeks have gone by.

  • So the cells as a function of the weeks of t,

  • which is in weeks, is going to be our original amount,

  • and then however many weeks have gone by,

  • we're going to multiply by 2/3 that many times,

  • so times 2/3 to the t power.

  • And we're done.

- [Instructor] We are told a phone sells for $600

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指數衰落單詞問題 (Exponential decay word problems)

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    林宜悉 發佈於 2021 年 01 月 14 日
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