Question

One model of Earth's population growth is
, where t is measured in years since 1990, and P is measured in billions of people. Which of the following statements are true?

Check all that apply.

A. In 1990, there were 5.33 billion people.
B. The population of Earth will grow exponentially for a while but then start to slow down its growth.
C. The population of Earth is increasing by a steady rate of 8% per year.
D. The carrying capacity of Earth is 5.33 billion people.

Answer 1

A: Have you considered substituting t = 0 into the model?
B: Anything that says "for a while" I have to find suspicious.
C: Well, that doesn't make sense.
D: What is the limit of P, as t increases?

Answer 2

I believe one answer is A and the other C

Answer 3

Test it out.

What is \(\dfrac{P(1)}{P(0)}\)? Is it 1.08?

What is \(\dfrac{P(2)}{P(1)}\)? Is it 1.08?

Answer 4

I can't figure it out

Answer 5

Did you evaluate P(0)? \(P(0) = \dfrac{64}{1+11\cdot e^{0.08\cdot 0}} = \dfrac{64}{1+11} = 5.333\)

Okay, you do P(1).

Answer 6

* Sorry, that exponent should have been negative. fortunately, for t = 0, it didn't matter.

Answer 7

p(1) = 4.95503

Answer 8

You didn't heed my apology. That's -0.08t, NOT 0.08t. Change that exponent to a negative and you'll have it. It didn't make a different for t = 1, but it does make a difference for ANY other value of t.

Answer 9

5.73771

Answer 10

That's it. What is the Percent Growth in the first year?

Answer 11

I'm not sure

Answer 12

@tkhunny

Answer 13

8%?

Answer 14

If you round to a whole percent, yes. Don't do that rounding.

Do the same for P(2)/P(1). If they are not the SAME, then it is not constant exponential growth.

Answer 15

5.73771

Answer 16

P(0) = 5.333
P(1) = 5.738
P(2) = 6.170

1st year percent growth: 5.738/5.333 - 1 = 7.6%
2nd year percent growth: 6.170/5.738 - 1 = 7.5%

There it is. The growth rate is not constant. It is slowing. Let's try another...

10th year percent growth: 10.770/10.072 - 1 = 6.9%

Yup. Still slowing.

On the other hand, it doesn't slow very much. It could be construed as Exponential Growth for a little while. It is definitely not exponential growth after that vague "little while".

100th year percent growth: 63.765/63.745 - 1 = 0.0306%

Answer 17

So would it only be A?

Answer 18

A. In 1990, there were 5.33 billion people.

What is P(0)? If it is 5.33, there we're good.

B. The population of Earth will grow exponentially for a while but then start to slow down its growth.

I think we just demonstrated that this IS so. It doesn't fit the strictest definition of "exponential", but it may be close enough for this model.

C. The population of Earth is increasing by a steady rate of 8% per year.

We disproved that.

D. The carrying capacity of Earth is 5.33 billion people.

The limit of P(t) is NOT 5.33, it is 64. So, no on this one.

Answer 19

Ok A is 5.33.

Answer 20

I'd go with A and B.

Answer 21

Thank you so much!