Please help! Confusing word problem!

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metrocknroll

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I have no idea how to do this one:

If a ball is thrown vertically upward with an initial velocity of 80 ft/sec., the distance s, in feet, of the ball from the ground, after t seconds, is given by s=80t - 16t^2. For what time interval is the ball less than 64 feet above the ground?
 
If a ball is thrown vertically upward with an initial velocity of 80 ft/sec., the distance s, in feet, of the ball from the ground, after t seconds, is given by s=80t - 16t^2. For what time interval is the ball less than 64 feet above the ground?

From s = 80t - 16t^2, 64 = 80t - 16t^3 or t^2 - 5t + 4 = 0.

This factors to (x - 1)(x - 4)

If t = 1, s = 80(1) - 16(1)^2 = 64

From Vf = Vo - 32t, Vf = 80 - 32t = 0 making t = 2.5 seconds for the ball to reach zero vertical velocity. Total flight time up and down is therefore 5 seconds.


Therefore, the ball is below 64 ft. at t = 1 sec. on the way up and below 64 ft. after falling for 1.5 seconds on the way down for a total of 5 seconds.

If you were looking for the time on the way up only, it would be 1 second.
 
The question asked for the entire time the ball is less than 64ft above the ground.

Using the quadtratic, as above, you can find that the two times the ball hits 64ft above the ground are 1 second and 4 seconds. So the total time above 64ft is 4-1=3 seconds.

The total flight time is 5 seconds as figured above, but the total time the ball is below 64ft is given by the total flight time minus the time it was above 64ft. So, 5-3=2 seconds.
 
My apologies for a slip in the numbers.

If a ball is thrown vertically upward with an initial velocity of 80 ft/sec., the distance s, in feet, of the ball from the ground, after t seconds, is given by s=80t - 16t^2. For what time interval is the ball less than 64 feet above the ground?

From s = 80t - 16t^2, 64 = 80t - 16t^3 or t^2 - 5t + 4 = 0.

This factors to (x - 1)(x - 4)

If t = 1, s = 80(1) - 16(1)^2 = 64

From Vf = Vo - 32t, Vf = 80 - 32t = 0 making t = 2.5 seconds for the ball to reach zero vertical velocity. Total flight time up and down is therefore 5 seconds.


Therefore, the ball is below 64 ft. at t = 1 sec. on the way up and below 64 ft. after falling for 1.5 seconds from its peak altitude, or for 1 second more, on the way down for a total of 2 seconds.

If you were looking for the time on the way up only, it would be 1 second.
 
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