@Duane: To keep ourselves amused while picking, we yelled back and forth between trees to collaboratively use our calculus knowledge to figure out how fast we'd be going when we hit the ground from that height.
You certainly knew how to have fun in those days :-)
Let's try again. first is it an aneroid or mecury barometer? If mecury then it will have a scale. Use that scale as the basis of developing a measuring stick or string. Then use the string to produce a 45 degree right angle triangle, along the top of a table and the height of the string. Then sight the top of the building along the hypotenuse of the triangle. Knowing the distance away from the building you can calculate the height of the tall building. If the building is very tall then it is easy to make both 45 and 60 degree triangles and use them to sight the top of the tall building. Now knowing the distance apart of the two sighting positions you can calculate the height of the building and the distance you are away from it.
If it is an aneroid barometer then make a simple parachute and drop the barometer attached to it from the top of the tall bulding. The barometer and parachute will quickly reach terminal velocity, so have a colleague a couple of floors of easily measurable distance down measure the time it takes to pass him/her and also the total time to reach the ground. The calculation of height is then trivial. This method avoids the problem of the calculation when throwing just the barometer off of the very tall building and a period of gravitational acceleration and then terminal velocity are involved.
Max re: "But that doesn't involve using the barometer, which is a key requirement for the exercise."
Convince the records office clerk that the barometer is actually a very valuable antique watch. Then bribe him or her with the watch to go right away and get the blueprints without delay, so you'll have enough time to stop at a barometer store and buy a new one on the way back to the building site.
Max, re: "Drop the barometer off the top of the building, measure how long it takes to hit the ground, and use this value to calculate the height of the building."
Many long years ago, I had a summer job working in the glorious Pacific Northwest forests. Part of the job involved climbing fir trees to pick the cones, which would be used in research plantings.
One day a buddy and myself were each in old growth Douglas fir trees, up in excess of 200 feet. To keep ourselves amused while picking, we yelled back and forth between trees to collaboratively use our calculus knowledge to figure out how fast we'd be going when we hit the ground from that height.