beanstalk doubling every hour what would speed of inchworms need to be?

Question

A beanstalk is continuously growing at a constant rate so that after one hour its length is doubled, after two hours it is four times the original length, after three hours it is eight times as long, and so one. (The growth is uniform: every part of the stalk grows at the same rate.) Two bugs start at a distance of one foot away from each other on the beanstalk and begin crawling towards each other at the constant speed x ft/hr (x is some number). For what values of x will the bugs meet?

Answer 1

Since both inchworms are travelling at the same speed they will meet at a point that is 0.5 feet between them. So I only have to consider one inchworm trying to get to a point 0.5 feet away.

The beanstalk grows according to $ l_b = 2^t $. So the location of our inchworm over time will be $ l_i = 0.5 + 2^t - xt $. We wish the worm to reach the middle so we can change that to being $ 0 = 0.5 + 2^t - xt $. Rearranging $ x = \frac{0.5 + 2^t}{t} $. This tells us how fast our worm must travel in order to meet up at time t.

I then used wolfram alpha to work out that the minimum value for x would be 2.20488.