Precalc question

Question

In 1987, the population of Mexico was estimated at 82 million people, with an annual growth rate of 2.5%. The 1987 population of the United States was estimated at 244 million with an annual growth rate of 0.7 %. Assume that both populations are growing exponentially.

(a) When will Mexico double its 1987 population?

(b) When will the United States and Mexico have the same population?

Answer 1

For exponential growth, the formula is:

P = a(b)^t where b = 1 + r and t is in years.

(For simplicity, we are going to divide the populations by a factor of 1000000. For instance, 82 million will be abbreviated as 82, and 244 million will be abbreviated as 244)

Part a

Let a = 82, b = 1 + .025 = 1.025, and P = 164, so our equation is:

164 = 82(1.025)^t

Divide both sides by 82 to get:

2 = (1.025)^t

Now, what you can do is take ln() of both sides to get the t out of the exponent. When we do this the exponent moves in front of the term on the right:

ln(2) = t * ln(1.025)

Finally divide by ln(1.025) on both sides:

ln(2)/ln(1.025) = t

t ≈ 28 years

For part (b)

Take the left hand side of the equation we created for Mexico in part(a) and set it equal to the equation we get for the population of the United States with the following information:

a = 244, b =1 + .007 = 1.007. Thus P = 244(1.007)^t

Setting them equal gives us:

82(1.025)^t = 244(1.007)^t

Divide both sides by 82 gives us:

(1.025)^t = 244/82(1.007)^t

Now, divide both sides by 1.007^t

(1.025)^t / (1.007)^t = 122/41 [if we reduce 244/82]

Applying the law of exponents for division: (a^x) / (b^x) = (a/b)^x gives us:

(1.025/1.007)^t = 122/41

We can apply ln() on both sides to get the t out of the exponent like we did in part (a):

t * ln(1.025/1.007) = ln(122/41)

Dividing both sides by ln(1.025/1.007) will give you your value of t:

t = ln(122/41)/ln(1.025/1.007)

t ≈ 65.5 years