Problem 50 Hard Difficulty

If the birth rate of a population is $ b(t) = 2200 e^{0.024t} $ people per year and the death rate is $ d(t) = 1460 e^{0.018t} $ people per year, find the area between these curves for $ 0 \le t \le 10 $. What does this area represent?

Answer

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Video Transcript

Okay, let's look at this problem. So here I have two curves. Be attendee of T B e r t represents the birth rate on DH the Opti the death rate. Okay, so we want to find the every had between the two cups for Teo term zero and ten So graphically So here I have B m t which looks like this. And then I have deal t which looks like that. All right. And here have zero and you have ten. So the area between the two curves which is this part we want to find So we all notice from the air represented two cars so we used a definitely to grow. So here let's set up the integral Teo integrate the difference between the two cuffs. So here be obvious Obviously greater than D of t. So we used to be a teacher. Subtract deal T which is two thousand two hundred he to the zero point zero to forty minus one four six zero it to the zero point zero one eight t t t And for the upper and lower down a bit agro which should be this tea goes from zero to ten okay. And then we just evaluated this stuff, innit? Integral. So this is equal to two thousand two hundred over the coefficient for the exponents. Zero point zero two four times E to the zero point zero two four two minus one four six zero over zero point zero one eight times. Need to the zero point zero one eight t and the party goes from zero to ten. So next is plugging numbers. Are you plugging tears? Even ten. That's twenty two hundred off zero point zero two four e to the zero point now. Two years. Ten. So this becomes zero point two four minus one for six. Zero over zero point zero one eight. Uh, E t. Now you have people to attend. So this exponents is zero point one eight, right? And then we subtract twenty zero to zero. Know that when t zero to zero, we have what? Eating zero here, Andy. To the zero here. We know that each of the zeros they would one. So when you plug into zero zero, you simply have twenty two hundred over zero point zero two four minus one for six. Zero over zero point zero one eight. And now we just use a calculator to find the number should be approximately. Ah, nine three three. Nine nine thousand three hundred thirty nine. Okay, so next question, what does this number mean? So we did. Here is we find the area between the two camps. So we're integrating the birth rate. Right? So if you just integrate, be up tea from zero to ten, this will be what This will be the number ofthe birth from zero to zero. T ten. If you just integrate D up from zero to ten will be the number of death. Um, between t zero zero ng t zero ten. So it is tracked. Those birth minus staff. That's the what is the increase in population. Right? So this number represents the increase in population from T is he goes zero to t zero to ten.