An oil company discovered an oil reserve of 130 million...

An oil company discovered an oil reserve of 130 million barrels. For time t > 0, in years, the company's extraction plan is a linear declining function of time as follows: q(t) = a - bt, where q(t) is the rate of extraction of oil in millions of barrels per year at time t and b = 0.15 and a = 12. (a) How long does it take to exhaust the entire reserve? (b) The oil price is a constant $45 per barrel, the extraction cost per barrel is a constant $14, and the market interest rate is 12% per year, compounded continuously. What is the present value of the company's profit?

Transcript

00:02 Hi there, so for this problem, we are told that an oil company discovered an oil reserve of 130 million barrels for a times t for deterrent or zero in years.

00:14 The company's instruction plan is a linear decline function of the following way.

00:19 So that will be q of d is equal to a minus b times the time.

00:25 Now we are given the value for b that is equal to 0 .15 and the value of a that is 3 .15 and the value of a that is so with this we can write that q of t is equal to then to 12 minus 0 .15 times the time.

00:43 Now with that said, for part a of this problem, the question is, how long does it take to assouse the entire reserve? so we have this rate that we are given.

00:54 So we aim to find the time that it takes to assouse the entire reserve.

00:59 So let the time taken to assounce the entire reserves to be 10.

01:03 So initially there was 130 million of powers, okay? so we're going to take the integral from zero to the times the expression that we're given for this, which is this one right here.

01:19 Let me just move it to here.

01:22 And then we integrate this over the time.

01:25 And then we know that this is equal to 130.

01:28 Now let's solve this different, this integral, okay? so we will have 12 times the time.

01:33 Minus 0 .15 divided by 2 times the time squared.

01:39 This is equal to 130.

01:41 So now we just need to solve for the time in here.

01:49 So what we can do is to move this term.

01:53 Well, we can move all of these two terms to the right.

01:58 So we will have 0 is equal to 0 .15 divided by 2.

02:13 This times the time squared, this minus 12 times the time, this plus 130.

02:23 Once we have this, you can recognize that this is a quadratic, yes, it is a quadratic equation, so we can use the quadratic formula to solve for the time that give us a positive value.

02:47 So first of all, let's, well, if we take the value, will we obtain two solutions for this? when the time is equal to 11 .69, and the other time is 148 .31, okay? so those are the two solutions.

03:10 Okay.

03:11 So now we need to verify if these solutions give us a negative value for q or a, a positive value for q...

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