Give Me a Sign

Posted on May 16, 2013 by laura

Billboards are those big rectangular signs along the side of the highway, usually trying to talk you into buying something. Since they’re a good 50 feet above you, these signs are even bigger than they look – they’re 20, 30, or even over 40 feet wide. The size depends on the speed of the traffic: the faster you’re driving, the easier it has to be for people to read it, so the bigger the letters and pictures need to be. The signs along slower streets in town, called “posters,” are 22 feet wide, but the big “bulletins” along the highway are up to 48 feet wide. Now we have digital billboards that light up like a computer screen and change the picture every few seconds, showing drivers even more things they should buy. Either one works, as long as we read it quickly: the more important thing as a driver is to keep your eyes on the road.

Wee ones: Which one is wider, a 20-foot-wide poster or a 40-foot-wide billboard?

Little kids: If a billboard is 10 feet tall and the bottom is 50 feet off the ground, how many more of those same signs could you stack edge to edge below it? Bonus: If you want to climb up to that 50-foot sign but all you have is an 8-foot ladder, how many more feet does your ladder have to reach?

Big kids: If an adult’s face is 9 inches tall, and all objects on a billboard are 12 times life size, how tall is a face shown on a billboard? Bonus: These giant highway signs are about as big as a house. If the front of a house has a 15-foot-wide kitchen, a 6-foot-wide hallway, and an 18-foot-wide living room, which one is wider, the house or a 48-foot-wide highway billboard?

The sky’s the limit: Suppose as a prank you decide to climb up a billboard at night and wrap a string of lights around the edge. If you need exactly 82 feet of strung lights to cover all four sides, and the area of the billboard (length times width) is 400, what are the width and height of the billboard?

Answers:
Wee ones: The billboard is the bigger number.

Little kids: 5 more of those signs. Bonus: 42 more feet.

Big kids: 108 inches, or 9 feet! Bonus: The billboard would be wider if leaned up against the house – 48 feet vs. 39.

The sky’s the limit: It’s 25 by 16 feet. We know that the width and height have to add to 41, since those two sides will use up half the lights. Those same two numbers also multiply out to 400. You can use trial and error to test the factors of 400 numbers: 40 by 10 doesn’t work, nor does 20 by 20, but 16 by 25 does. Expressing this using algebra:
w + h = 41, so h=41-w
w x h = 400. Replacing h, you get
w x (41-w)=400
41w – w^2=400, or w^2-41w+400=0
…and then you still need trial and error to break it down into
(w-25) x (w-16)=0. So w=16 or 25.

Digging Out

Posted on May 10, 2013 by laura

As long as we’re talking about vehicles and water, this news story might top the tugboats. A group of travelers in Siberia reached a flooded river that they had to cross, so they got creative. They used two excavators – those earth-diggers with the giant scoop-shaped shovel at one end – to carry everyone across. As you can see in this video, a handful of people at a time climbed into the shovel of one excavator, which then swung its arm out over the river. The other excavator swung its shovel out to meet the first shovel, and the riders climbed from one shovel to the other, handing over luggage as they did. Then the next group climbed in. This can’t be the fastest or easiest way to travel, but at least no one fell in the river – the water in Siberia is pretty darn cold!

Wee ones: If the shovel can hold 8 people and you’re 6th in line, will you get to climb in with the first group?

Little kids: If 8 people can cross the river in the shovel and half those people have luggage, but of the people who have luggage, half have 2 suitcases and the others have just 1, how many suitcases have to fit in the shovel? Bonus: If it takes 10 seconds for each person to climb in, 11 seconds to switch shovels, and 8 seconds to climb out, how many seconds of climbing does each passenger have to do without falling into the river?

Big kids: That looks like a long line of people waiting to cross…If the shovel holds 8 people and there are 59 people who want to cross, what’s the minimum number of groups the excavators have to ferry across? Bonus: If another 22 more people show up before they’ve finished, now how many trips will it take – and can you solve that quickly without adding 59 and 22?

The sky’s the limit: If each excavator can swing between shore and mid-river in 5 seconds (whether empty or full), and climbing in takes 10 seconds, the mid-river hand-off takes 20 seconds, and climbing out takes 15 seconds, how long does it take for the whole group of 59 to cross the river IF the first excavator always returns to load the next group of 8 immediately?

Answers:
Wee ones: Yes! And two more can climb in after you.

Little kids: 6 suitcases. 4 people have luggage, and 2 of them have 2 suitcases each while 2 of them have 1 apiece, making 2+4. Bonus: 29 seconds of climbing.

Big kids: 8 trips, since the first 7 trips will take just 56 people. Bonus: It will now take 11 trips. The 22 people will fill 2 full trips and have 6 folks left over. 5 of those will fill the 8th trip from the original group, then one poor guy will have to ride alone.

The sky’s the limit: 370 seconds (or 6 minutes 10 seconds). The first group takes 10+5+20+5+15, or 55 seconds to get across. However, each group after that adds a little less time, because when the second excavator swings back to mid-river the 1st excavator is already waiting. The 2nd excavator needs 5 seconds to swing back, 20 seconds to collect people, 5 to get to shore, and 15 to unload. So you add only 45 seconds for each of the next 7 groups. That makes the total 55 + seven 45′s, or 370 seconds.

Work Hard, Play Hard

Posted on May 9, 2013 by laura

You can tell from their name what tugboats do: they tug other boats. Tugboats move ships that can’t move themselves, like heavy barges, and steer big ships through crowded areas so they don’t crash into each other. Tugboats can also act as icebreakers, help fight fires, and rescue sinking ships. And to top it off, they can do dance routines: today the annual “Tugboat Ballet” kicks off in Germany, in which eight tugboats swim around to waltz music blasted over loudspeakers, to show off how zippy they are. They also show off their speediness in tugboat races in cities around the world, and the New York race also features a nose-to-nose pushing contest and a line toss competition. Tugboats may not be the biggest boats out there, but it’s clear they have the most fun.

Wee ones (counting on fingers): If 2 of the 8 tugboats scheduled to take part in the Tugboat Ballet break down at the last minute, how many tugboats can still do the show?

Little kids: In the nose-to-nose pushing contest, if one boat pushes the other back 14 feet, but then that boat pushes the first boat 21 feet from there, how far from the starting point do they end up? Bonus: A tugboat can push up to 40 barges, all way bigger than itself! If each barge weighs 5 times as much as the tug, how many times its weight does a tug pulling 20 barges tow?

Big kids: In the New York City tugboat race, the tugs start next to 79th Street in Manhattan and end at a pier next to 44th Street. How many blocks do they travel? Bonus: In last year’s race, the first boat crossed the finish line in 5 minutes and 1 second, and the last one in 12 minutes and 48 seconds. How many total seconds passed between the first boat’s finish and the last?

Answers:
Wee ones: 6 tugboats.

Little kids: 7 feet in front of it. Bonus: 100 times.

Big kids: 35 blocks. Bonus: 467 seconds (7 minutes 47 seconds).

Car Wash of Doom, and Getting in Gear

Posted on April 23, 2013 by laura

It’s another two-fer Tuesday – and that means it’s time for “Car Wash of Doom,” our fourth Bedtime Math video. In this episode, our hero Kevin tries to work at a car wash, but as you can guess from the title, it doesn’t go so well. Click here to watch! Just one more way to get jazzed about math here at Bedtime Math.

Of course, as a kid you can’t drive a real car, but there are toy ones that drive themselves just as well. In a wind-up car, the little handle you twist turns a gear inside that’s connected to a spring. When you let go, the spring wants to snap back, which makes the gear turn, which makes all the wheels on the floor turn. A pullback car uses the same idea: you just roll the car backward, and the car wheels do the job of turning that gear. And how come just a few twists make the wheels turn so many times? Because when you turn a big gear with lots of teeth that’s touching a small gear with very few teeth, one turn of that big gear makes the little gear turn many times – and makes your car harder to run after.

Wee ones: Which one has more wheels on the floor, a toy car with 4 wheels or a truck with 6 wheels?

Little kids: If each twist of the wind-up handle makes the wheels roll 5 times, how many times will the wheels turn after 2 twists? Bonus: How about 4 twists?

Big kids: If a big gear with 48 teeth turns an interlocking little gear with only 6 teeth, how many times will the little gear go round for each turn of the big gear? Bonus: If the axle with the little gear has its own 48-tooth gear turning another 6-tooth gear, how many times will that last little gear turn for each turn of the first big gear?

Answers:
Wee ones: The 6-wheeled truck has more.

Little kids: 10 turns. Bonus: 20 turns.

Big kids: 8 times. Bonus: 64 times, because the second big gear will turn 8 times and each of those turns will make the last gear turn 8 times itself.

Space Chase

Posted on April 21, 2013 by laura

It’s one thing to dig around in the backyard looking for cool rocks. It’s a whole other story to send a rocket into outer space to grab asteroids. That’s exactly what America’s space agency NASA wants to do in its Asteroid Capture and Return mission. The idea is for a spacecraft to fly millions of miles to the Asteroid Belt — the ring of rocks that orbit between Mars and Jupiter — capture a 25-foot rock, bring it back, and drop it into the same orbit as our Moon, so astronauts can catch it and study it. Not only do asteroids hold clues about our solar system, but they might also have useful materials in them, like iron and titanium…if this crazy plan works, we could mine asteroids for metals. A lot more work than hunting around the backyard, but a lot more magical, too.

Wee ones: If you fly into space and catch every other asteroid you see, and so far you’ve caught the 1st, 3rd and 5th asteroids you’ve seen, will you catch the 8th one?

Little kids: Asteroids orbit the sun in a rough circle between Mars and Jupiter. Mars is closer to the Sun than Jupiter…if one of those orbits is about 140 million miles from the sun and one is about 780 million miles, which one is Mars’ orbit? Bonus: If the project starts in 2014 and the rocket is ready for launch 5 years later, in what year will the rocket launch?

Big kids: The deadline for the mission to be done is 2025. If starting in 2014 the craft takes just 2 years to build, then once launched the craft takes 1 year longer to reach the asteroid as to come back, in what year does the spacecraft catch the asteroid? Bonus: If the asteroid we want is exactly halfway between Mars’ and Jupiter’s orbits, how far from the Sun is that? (Again, Mars averages about 140 million miles out, and Jupiter about 780 million miles.)

The sky’s the limit, literally: If Earth and our target asteroid are right now as close together as possible in their orbits, and the asteroid takes 7 Earth-years to orbit the Sun in the same direction as Earth, how many months from now will they again be as close together as possible?

Answers:

Wee ones: No – you’ll catch the 7th and 9th instead.

Little kids: Mars is the 140 million mile orbit. Bonus: In 2019.

Big kids: In 2021. The craft will be ready in 2016, leaving 9 years for the trip, so the first leg will take 5 years to reach the asteroid (vs. 4 to return). Bonus: 460 million miles.

The sky’s the limit: 1 1/6 years, or 14 months. Earth will pass the asteroid 6 equally spaced times over the 7 years, with the 6th happening at their starting points. So, although the asteroid will have done 1/7 of its trip when Earth first passes the original starting point, it will be 1/6 of the way around by the time Earth passes the rock.