# How is the time doubling formula used?

I missed several days of my math class and now have no idea what is going on. We're working on Real Population Growth. I tried to figure this problem out myself, but apparently I'm not getting one of the values right because when I check my answer, it isn't correct. The problem is as follows:

Starting from an estimated U.S. population of 305 million in 2009, use the given growth rate to estimate U.S. population in 2059 and 2109. Use the approximate doubling time formula.

The growth rate they gave me is 0.5% for this problem. I set it up like:

Tdoubling = 70/0.5 = 140

(305 * 10^6) * 2^50/140

50 for the 50 years from 2009 to 2059 and I don't think 140 is the correct value but I don't know what else to try.

The correct answer is 391 million for 2059, I just can't figure out how to arrive at it.

• What is your approximate doubling time formula? How many years does it take something to double at $0/5\%$ per year increase? We can't say what is wrong if we don't see your work. – Ross Millikan Apr 8 '19 at 15:34
• @RossMillikan I used 70/0.5... I got 140 from that so I would assume for 2059 would be 50/140 but I'm not getting the right answer so I'm not sure what's going on. Thanks for letting me know I missed some work. – Amanda Apr 9 '19 at 2:36
• I suspect you did not evaluate (305 * 10^6) * 2^50/140 the way you intended to. It should round to $391$ million. Your expression does not have the $140$ in the exponent. Maybe your calculator followed the proper order of operations. – Ross Millikan Apr 9 '19 at 2:55

The correct calculation for $$0.5\%$$ growth over $$50$$ years from $$305$$ is $$305\cdot 1.005^{50}\approx 391.38$$. Your expression for the doubling formula is $$305 \cdot 2^{(50/140)}$$ [note the parentheses-the way you wrote it is $$(305 \cdot 2^{50})/140$$, very different]. This is about $$390.67$$, which rounds to $$391$$ as well.