SOLUTION: I am really having troubles with these word problems. the equation -16t^2 - 32t + 400 gives the height of a ball, thrown downward from the top of a 400 ft building with an ini

Algebra ->  Radicals -> SOLUTION: I am really having troubles with these word problems. the equation -16t^2 - 32t + 400 gives the height of a ball, thrown downward from the top of a 400 ft building with an ini      Log On


   

Question 762491: I am really having troubles with these word problems.

the equation -16t^2 - 32t + 400 gives the height of a ball, thrown downward from the top of a 400 ft building with an initial velocity of 32 ft/s. find the time it takes for the ball to reach a height of 96 ft
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+h%28t%29+=+-16t%5E2+-+32t+%2B+400+
First of all, I'll show you the plot of this equation:
+graph%28+400%2C+400%2C+-2%2C+6%2C+-50%2C+450%2C+-16x%5E2+-+32x+%2B+400+%29+
You can see the height of the building at +t+=+0+
+h%280%29+=+-16%280%29%5E2+-+32%2A0+%2B+400+
+h%280%29+=+400+
So, the height of the ball is the same as the height of the building
---------------------
The problem doesn't ask you for it, but you can also see that
the ball hits the ground at about +t+=+4.1+ sec or so.
The height is zero, so +h%28t%29+=+0+
Solving:
+h%28t%29+=+-16t%5E2+-+32t+%2B+400+
+0+=+-16t%5E2+-+32t+%2B+400+
+-t%5E2+-+2t+%2B+25+=+0+
Using quadratic formula:
+t+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+-1+
+b+=+-2+
+c+=+25+

+t+=+%28+2+%2B-+sqrt%28+4+%2B+100+%29%29+%2F+%28-2%29+
+t=+%28+2+-+sqrt%28+104+%29+%29+%2F+%28+-2%29+
+t=+%28+2+-+10.198+%29+%2F+%28+-2%29+
+t+=+%28+-8.198+%29+%2F+%28-2%29+
+t+=+4.099+
My guess was pretty good
----------------------
They are asking you for +t+ when +h%28t%29+=+96+
+h%28t%29+=+-16t%5E2+-+32t+%2B+400+
+96+=+-16t%5E2+-+32t+%2B+400+
+-16t%5E2+-+32t+%2B+304+=+0+
+-4t%5E2+-+8t+%2B+76+=+0+
+-t%5E2+-+2t+%2B+19+=+0+
Use quadratic formula
+t+=+%28-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+-1+
+b+=+-2+
+c+=+19+

+t+=+%28+2+%2B-+sqrt%28+4+%2B+76+%29%29+%2F+%28-2%29+
+t+=+%28+2+%2B-+sqrt%28+80+%29%29+%2F+%28-2%29+
+t+=+%28+2+%2B-+sqrt%28+16%2A5+%29%29+%2F+%28-2%29+
+t+=+%28+2+-+4%2Asqrt%28+5+%29+%29+%2F+%28-2%29+
+t+=+2%2Asqrt%285%29+-+1+
+t+=+3.472+ sec
This seems to agree with the plot
-----------------------------
Just for explanation, suppose the +-32t+ had been +
+h%28t%29+=+-16t%5E2+%2B+32t+%2B+400+
Here's the plot:
+graph%28+400%2C+400%2C+-2%2C+7%2C+-50%2C+500%2C+-16x%5E2+%2B+32x+%2B+400+%29+
This is the case where the ball is thrown upward instead of down.
Hope all this helps