## Friday, May 18, 2012

### How Tall is Your House?

Imagine that you are shopping for a house and a Real Estate agent takes you to an amazing contemporary that looks something like this:

You inspect the house and climb up on top of the roof. You would like to know how tall the house is because a new law allows owners in this area to add another floor if the height of the house is below 10 meters. Unfortunately, blueprints with the specified height couldn't be found. Are there any experiments you can do to measure the height of the house? There may be more than one answer.

Your answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.

Magic Kid said...

2quick solutions:
Get a rope of 11m(that is height threshold so after that it does matter.
since you can get on top of roof and connect via tape or knor to roof so 10 m is left to drop, drop down and measure leftover after it touches ground and subtract from 10.
Drop a ball from roof height(since you're up there or if you want to keep standing roof height plus 1 meter. Time with a good stopwatch until it hits ground. height=1/2*(9.81)(time)^2
or of you are 1 meter up height+1=1/2*(9.81)(time)^2. Where time is the number of seconds measured when it was dropped.

There are also a few other ways to do this fairly easily.

Lynnet's mom said...

This reminds me of the classic puzzle about using a barometer to measure the height of a building (with answers including "measure the height of the barometer and the length of barometer and building's shadows and calculate building height via similar triangles," "drop the barometer off the side and time how long it takes to smash on the ground" (which would not work here because it's too short to time accurately) and "find the superintendent of the building and offer him the barometer as a gift if he can tell you how tall the building is").

My favorite includes going to the roof and using a string and a weight (say, the barometer) to get to the ground. Sure, you could just measure the length of the string. But it might stretch. Better to use the string as a pendulum and measure its period, which you could then use to get the length of the possibly-stretched string.

Anonymous said...

A few ideas for measuring the height of the house:

(1) Drop a large ball (100ft+) of heavy string cord off roof, hanging onto the end of the cord and then measure how much cord came unwound during the fall.

(2) If it is sunny, and the sun is in the right part of the sky so that the shadows fall on the driveway, estimate the height using shadows and proportional triangles. Take a stick or pole of known height (h1) and measure the length of its shadow (s1) on the ground. Then measure the length of the shadow of the house (s2), and then calculate h2 as h1*s2/s1.

(3) Drop a medium sized rock off the roof and measure how long it takes to fall to the ground and then calculate the distance as 1/2*a*t*t, when a=acceleration (9.8m/s^2), and t=time to fall in seconds. If the fall is recorded using a mobile video app, then the video could be played back frame by frame (many programs record at 30 Hz) to more accurately estimate the time.

(4)... My last suggestion, and this is the least interesting option imho would be to stand on the roof and use a contractor's measuring tape to measure the height. Contractor's measuring tapes often go out to 100 feet, sometimes more. 100 feet converts to 30.48 meters, so if the person measuing from the roof starts exactly at the edge of the top of the roof, the tape would be sufficient measure if the roof is less than 30 meters or more.

TracyZ

anne-marie said...

This person could put a stick ( with known dimension k) up aligned with the ground point and the high point and use Thales theorem. The dimension from the endpoint tangent of the triangle to the stick l could be measured. The tangent being the line joining the top of the house and going through the stick to the ground.
The distance from the house to the stick m could be measured so Thales theorem could be applied to find the height h of the house.
h/(l+m)= k/m

Jerome said...

I can think of three methods:

We live in the country and always carry rope. Most ropes are sold in bundles of 25 to 50 feet in length. If you can climb up on a roof then all you need do is put a plumb on the end (I always have a Swiss Army knife on my keychain. Don't all "real Men" and “real women” have the same tool on them?) and drop it down.

Most of the time, I also have a measuring tape, but if I'm not allowed that then I can measure it against the width of the lot. If all I'm interested in the building’s height (7m) then a little more than twice the height of the building will be the width of standard lot in most places. (Most lots are 50 feet wide in North America). Seven meters equals 23 feet about. Two times seven meters equals 46 feet, so two times the ropes length leaves you 4 feet short of the width of your lot. Any larger remainder and you are good to go to build another story.

Of course I could always measure the shadow of the building and compare it to the shadow of a vertical stick of known length.

Then there's the use of trig. Stand back from the building. Put a protractor on the ground. Use a pointer of (a straight branch on the ground) to determine the angle to the top of the building. Use

Tan (angle) = height of building/ distance between edge of building and the protractor.

The problem is how many people carry a protractor around in a back pocket??? Even real men or women don’t do that.

On a cloudy day methods 1 and 3 work best.

Jerome said...

I misread the height cutoff. I thought it was 7 meters but it turned out to be 10.

10 meters is about 32 feet. A standard city lot is 125 to 150 feet en length. If you measure the height of a building using a plumb then if the height is less than 4 lengths of the rope you can pretty much say you are good to go. Of course this is only an estimate.

Ryan said...

Drop a ball from a height level to the roof and record the time it takes to hit the ground in seconds. Square that number and multiply by 16. That's the height of the house in feet.
I'm using h(t)=-16t^2+vt+h where v is the initial velocity (zero if you're dropping the ball) and h is the initial height (the height of the house in this case). h(t)=0 when the ball hits the ground giving us:
0=-16t^2+h and with a little algebra:
h=16t^2 Dropping the ball gives t...

Annie said...

Hmmm...assuming I couldn't get my hands on a really long tape measure, I'd drop a rock from the roof and time how long it took for it to hit the ground. Using the following formula, you could calculate the height of the house.

Hgt. = 1/2 g(t2) where g=9.81m/sec/sec (gravity) and t = time

Also, I was wondering if you could measure the hgt. of your shadow and the hgt. of the house's shadow at the exact same time and at the exact same spot and do a ratio:

And maybe we could figure the hgt. of the house if we knew the length of the house's shadow and the angle of the sun when measuring the shadow.......

Jerome's Wife said...

House height – I can't solve the problem of determining the house height, except to throw in a few ideas. If one were to stand on top of the house and throw a nice straight rope with no kinks, down to the ground (with a plumb on it)...counting the feet of that rope would be a very close estimate of the height. To make measuring of the rope easier, one could reel it in on a reel. The reel could knick off each foot by some method of measuring device attached at the point where the rope enters into the axis of the reel, and then enumerating each foot on a meter or some counting device. To start with two feet would already be started on the reel, so the end measurement would simply be to add those 2 feet back into the final result. Or find out ahead of time, by trial and error, how many feet the reel would hold of rope of a certain dimension.

Or, add a second floor, by putting it under the first floor (since the house sits on stilts), and rename the first floor, the second floor. Bugs Bunny did it to Elmer, except in reverse. Elmer must NOT push the “wed” button, which was just too much of a temptation for Elmer. In this case, the second floor became the first floor. Elmer, was a bit miffed and was able to retrieve his hunting gun from the rubble to hunt down that “dwatted wabbit” who outwitted Elmer once again.

Dennis (of Dennis and Katrina) said...

THE ALGEBRAIC SOLUTION:

The distance (d) travelled by an object falling for time (t) can be expressed by:

d = 1/2(g)(t)^2

on earth, g = 9.8 m/sec^2

for the short distance/time the object will fall we can neglect air resistance.

Drop an object from the roof and time how long it takes to hit the ground. Square it and multiply it by 4.9 (1/2 * 9.8).

If you don't feel like doing the math, you can take advantage of the fact that the magic number is 10 meters and estimate by plugging 10 in for d:

10 = 1/2(9.8)t^2

9.8 is pretty close to 10. If you multiply both sides by 2 and divide by 10 to "remove" the 9.8, you get:

2 = t^2

t = 1.43

Knowuing that, drop an object. If it takes more than 1.43 seconds to hit the ground, you can't build. If it hits the ground in less than that time, you're okay. If you *REALLY* don't want to do the math, you can try:

THE GEOMETRIC SOLUTION:
(Assuming the ground is fairly level)

1) Find a stick the length of your arm.

2) Hold your arm out straight with the stick pointing straight up (90 degree angle to your outstretched arm).

3) Position yourself so that the top of the stick lines up with the top of the house.

4) Your feet are now at approximately the same distance from the house as it is high.

Lynnet said...

I have two ways.
1) Drop something off the roof of the house and measure the time it takes for it to hit the ground. Then calculate the distance. This requires you to be able to accurately measure the time.

Maria said...

Wow, wow, wow - how many solutions!

Let me try to summarize them all:

1)Hang a rope with something heavy at the end from the roof. Either measure the rope later or use the rope that is 10 meters long.

2)Drop something heavy and measure the number of seconds it takes to drop. Use the (Newton's?) formula to calculate the distance from time. It is a bit problematic as the threshold number of secs is 1.4 (as Dennis explained). If one is able to distinguish between 1.3 and 1.5 secs then it works.

3)Detect house's height from the proportion of your's and house's shadows. I don't think it is Thales' theorem but it was Thales who suggested to measure the height of Egyptian pyramids with this method. A very neat trick!

4)Measuring tape, contractors tape. But why would you have a 10 meter measuring tape in your purse? Rope and stone - I understand, but measuring tape? Just kidding...

5)If you have a protractor handy, Jerome suggested placing it on the ground and measuring height of imaginary triangle from the protractor's base to the base of the building and then it's top. Then, measure the distance from protractor to the building. You have one side, one angle and a right triangle. Find another side that is the height. Cool.

6)A very creative solution from Jerome's wife - build the second floor below the first one. There seems to be a space in the pic. Rename the floors. Apparently Bugs Bunny did it :)

7)Again use similar triangles (like in the shadow solution) but create them by using your body and a stick. See Dennis's solution.

I think we should give one puzzle point per two solutions, therefore everyone who proposed 1-2 solutions will get 1 puzzle point, 2-4 solution will get 2 puzzle points and more solutions will get 3 puzzle point.

Bravo everyone!

Maria said...

Lynnet's mom - could you please tell us more about the pendulum solution. Sounds very interesting!

Ilya said...

curious legend on the subject: http://www.snopes.com/college/exam/barometer.asp

Jerome. said...

The pendulum follows the formula

T = 2 * pi * sqrt(L / g)

This means that the time it takes to swing back and forth from the left most point to the right most point and back to the left most point. At 10 meters this is a lot of time. Let's just do it for 10 meters and see what we get.

T = 2*pi*sqrt(L / g)
L = 10 meters.
pi = 3.14 (two places are enough)
g is the gravitational constant on earth
T = 2*3.14 * sqrt(10/9.81)
T = 6.34 seconds.

Now the 0.34 seconds is hard to measure, but the 6 seconds is not so bad.

Of all the methods that don't use a measuring tape, this one is likely the best because the time is so long.

Maria said...

Thank you, Jerome, for the explanation.
This is a really good solution, and a feasible one - let's call it #8.

You all are a bunch of very creative people I want with me if I ever find myself on an uninhabited island.

Lynnet's mom said...

Sorry, I didn't check here earlier. Yes, Jerome's explanation is what I was thinking of. You can get the formula from a basic physics text. And yes, the reason I prefer that solution is that it gets the length of time you're timing to be long enough that you aren't running into significant-figures problems.

I don't think the drop-it-off-the-roof solution is going to be very accurate, unless you can use technological means to synchronize the stopwatch with the things you see/hear (automatically turn on at the moment of release, automatically turn off at the sound of hittting the ground, which will work in this circumstance but not for a very tall building because the relatively slow and highly variable speed of sound then would have to be taken into account.