00:01
So in this question, they say that a computer that is purchased today depreciates in value according to the function v of t equals 1 ,100e to the power of negative t over 3, where t represents time in years.
00:13
In part a, i want to know what was the purchase price of the computer.
00:19
Well, today is t equals 0.
00:24
So the value at that time is going to be 1100 times e to the power of 0.
00:31
E to the zero power is 1.
00:33
1 times 1100 is 1100.
00:36
And there's my answer to part a.
00:40
In b, we say, what was its value after one year? so now my t is 1.
00:48
I'm going to plug in and my v is equal to 1100, e to the power of negative 1 3rd.
00:57
I'm going to go to my calculator, 1 ,100, e to the power, 1 ,3rd.
01:05
Of negative one third.
01:07
And i get $788 and 20 cents.
01:11
They said round to the nearest decimal, to one decimal place.
01:15
So $788 and 20 cents.
01:20
In c, i want to know, how long will it take for the computer's value to decrease to half of its original value? well, half of 1100 is 550.
01:34
So i am setting 550, equal to 1100 e to the power of negative 2 over 3.
01:42
If i divide by 1100, i've got one -half equals e to the power of negative 2 over 3.
01:49
I'm gonna take the natural log of each side, so the natural log of 1 1 1 1ā2 equals negative t over 3.
01:59
I'll multiply both sides by negative 3, and i am getting t equals negative 3 times the natural log of a half, negative three times the natural log of one half, and we are getting 2 .1 years.
02:18
So it's going to take 2 .1 years to depreciate two half of its value.
02:25
And we say at what rate will the computer's value be depreciating at this time? well, now i need my v prime.
02:35
So my v prime, the derivative of 1 ,100 e to the power of negative 2 over 3.
02:42
That's 1100, e to the power of negative 2 over 3.
02:47
And then i'm going to multiply by negative 1 3.
02:51
And we are going to evaluate this at the time that we had here.
02:58
So we are going to evaluate this v prime at the place where t equals negative 3 times the ln of a half.
03:09
So what am i getting? well, let's see...