So I was going through this whole problem set and trying to solve everything I could without a calculator as a sort of self-imposed challenge. It was tough but somehow I'd managed it with all 32 problems... except the last one.
Let me quote it to you:
"A screwdriver is dropped from the top of an elevator shaft, exactly 5 seconds later the sound of the screwdriver hitting bottom of the shaft is heard. How deep is the shaft?
Hint: The distance that a dropped object falls in
t seconds is represented by the formula d=16t
2. The speed of sound is 1100ft/sec.
The final answer, spoilered for those of you that actually want to try solving this:
So the number of seconds elapsed is 5, this is equal to the time it took for it to hit the bottom plus the time it took for the sound to reach us.
Since the time it takes for the sound to go from the bottom of the shaft back to us is the distance d divided by 1100 we can write this all in terms of t, the time it took for it to reach the bottom of the shaft, giving us:
t+(16t2/1100)=5
Giving us our formula for the amount of time. Then we set it equal to zero...
t+(16t2/1100)-5=0
Rewrite it a little bit for clarity...
(16/1100)t2+t-5=0
The we use the quadratic formula to find our zeros:
(-1(+-)sqrt(1-4(16/1100)(-5)))/(2(5/1100))
I apologise for it being so messy, but this forum doesn't handle fractions very well, and I can't be bother to shove it into an image or something.
We're uninterested in a negative solution, so we can drop that negative sign and clear things up a little while we're at it.
(-1+sqrt(23115/1100))/(10/1100)
But of course this is just the time it took to reach the bottom of the shaft, we want the distance, which is expressed by the formula 16t2, so we plug that in and get:
16((-1+sqrt(24115/1100))/(10/1100))2
Needless to say, there was really no way I was going to work this out by hand. Especially since it wanted a decimal point rather than a fraction.