It isn’t every day that a major city gets a new bridge, but that’s what’s happening in the New York City area these days. They need a giant new bridge across the Hudson River because the Tappan Zee Bridge is getting so old. It’s been in use almost 10 years longer than it was supposed to last, and carries 138,000 cars per day when it was built to handle only 100,000. It would cost so much to keep fixing and strengthening the old bridge that at that point, the nearly $4 billion is better spent building a brand new one. The new bridge will have a 40-foot wide gap between the two directions of traffic in case the city wants to add a train track in between. Best of all, the new bridge will last at least 100 years. What we want to know is, how does the big old Tappan Zee feel about all this?
Wee ones: If it takes you 5 minutes to cross a bridge going one way, but 1 minute longer to come back thanks to more traffic, how long does the trip back take?
Little kids: Construction on the new bridge will finish in 2018 (or so they think). How many years from now will that be? (Reminder: We’re in 2014 right now.) Bonus: The new bridge is supposed to last 100 years! If it finishes in 2018, until when will it last?
Big kids: The new bridge’s northern span (set of lanes) will be 96 feet wide and the southern span will be 87 feet wide. Together with the 40-foot gap, how wide will the whole bridge be? Bonus: Highway lanes are at least 12 feet wide. If the northern span had 5 feet of extra space outside its lanes, and the southern span has 9 extra feet, what whole-number lane width would they be using, and how many lanes would each side have?
The sky’s the limit: If the current Tappan Zee carries 138,000 cars a day right now, and carries 1,000 more than that tomorrow, then 2,000 more than that number the next day after that, then 3,000 more than that new count the day after that…on what day will it first exceed 200,000 cars per day? (You can count tomorrow as day 1.)
Wee ones: 6 minutes.
Little kids: 4 years from now. Bonus: Until 2118.
Big kids: 223 feet. Bonus: At 91 and 78 feet, they would have to have 13-foot lanes, the only common factor. That’s 7 lanes for the northern span, 6 for the south.
The sky’s the limit: On the 11th day, at which point the bridge will be carrying 66,000 cars more than now. Adding, 1,000, 2,000, 3,000 etc. gives you 1,000 times the famed “triangle” series of numbers, which can be laid out as nice neat triangles. So we need the triangle number that will get us from 138 to 200, a gap of 62. The triangle numbers follow the formula x(x+1)/2: stacking 1 on top of 2 on top of a bottom row of 3 gives you 6 (half of 3×4), a bottom row of 4 gives you 10 (half of 4×5), and so on. You can continue like this until you reach 55, which isn’t enough, and 66, which means you’ll cross on the 11th day (half of 11×12). Alternatively, you can double 62 to get 124, then figure out the closest number with two factors 1 apart. 121 is 11×11, so you can narrow in on 11×12.
And a big, big thank-you to Rick H. for this great topic!