(2 points) An oil company discovered an oil reserve of 110...

(2 points) An oil company discovered an oil reserve of 110 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 $b = 0.05$ and $a = 14$ (a) How long does it take to exhaust the entire reserve? time = years (b) The oil price is a constant 45 dollars per barrel, the extraction cost per barrel is a constant 20 dollars, and the market interest rate is 11 percent per year, compounded continuously. What is the present value of the company's profit? value = millions of dollars

Transcript

00:05 Now in this question, we have all your component, right? they discovered all your reserve of this much barriers, right? and then the extraction rate is given by this equation, basically, and with a and b given by these numbers.

00:19 So you ask how long does take to exhaust the entire reserve? and obviously you just integrate qt, right? from the time now to the future t, which is one to find out, and time to integrate this function, you should say, to be 1 -1 -0, right? and then obviously if you do the integral, you obviously get from the a, you get a -t, right? and from b you get b, actually b over 2 and t squared, right? and this should be 1 -1 -0, right? and of course, this is a quadratic equation you can easily solve, right? so the quadratic equation, you can easily solve.

01:04 So basically use the formula, right, t.

01:07 Let me rewrite this quadratic equation in a lot of it.

01:10 So i will rewrite like b over 2, t squared, and minus a t and plus 110 being zero, right? and then, of course, the t, i can write it as the solution is simply given by 2 times a, right? that's b over 2 and times a plus.

01:34 A square root of a squared minus four times b over two and times one one ten right basically right so you plug in the values you should be able to easily work this out right so the a is 14 right so you'll find this to begin by actually 14 plus the square root of a squared that is 14 squared and then minus 2b, right, 2 times b, that's 0 .1 times 110, that's of course, 11.

02:15 And then the whole thing divided by 2, that divided by b basically, right, 0 .05.

02:21 And you'll find this trajectory about, wow, that's a non -time, right, is 552 years.

02:28 So it would take it this much years, right, 552 years, right? so what will be the company's profits, even if all your price is, $5 per barrier and extraction cost will be this much...

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