SOLUTION: A ball is thrown upward with an initial velocity of 50 feet per second. The height h (in feet) of the ball after t seconds is given by h=-16^2+50t. At the same time, a balloon is r

Algebra ->  Square-cubic-other-roots -> SOLUTION: A ball is thrown upward with an initial velocity of 50 feet per second. The height h (in feet) of the ball after t seconds is given by h=-16^2+50t. At the same time, a balloon is r      Log On


   

Question 1028733: A ball is thrown upward with an initial velocity of 50 feet per second. The height h (in feet) of the ball after t seconds is given by h=-16^2+50t. At the same time, a balloon is rising at a constant rate of 20 feet per second. Its height h in feet after seconds is given by h=20t.

a.) when do the ball and the balloon reach the same height?

b.) when does the ball reach its maximum height?

c.) when does the ball hit the ground?

Please help me, gladly appreciated <3
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There are to functions of time, and time is measured from the same moment taken at t=0 fro both functions. (That is not very clearly stated in the problem, but your teacher thinks it is understood to be that way).
The function h%28t%29=-16t%5E2%2B50t is a very realistic representation of what the height (in feet) of a ball would be t seconds after it is kicked up (or shot up) from the ground. (A physics teacher would like that equation).
The function h%28t%29=20t must represent the height (in feet) of a balloon t seconds after the ball is kicked/shot up. (I believe this is not very realistic for all t%3E=0 , and it would take some skill on the part of the balloon pilot to achieve that for some time).

a.) The ball and the balloon reach the same height when
20t=-16t%5E2%2B50t
16t%5E2-50t%2B20t=0
16t%5E2-30t=0
%2816t-30%29t=0
Obviously, that would be true for
system%28t=0%2C%22or%22%2C16t=30%29--->system%28t=0%2C%22or%22%2Ct=3%2F16=1.875%29
So, the ball and the balloon reach the same height "after highlight%281.875seconds%29".
That means 1.875 seconds after t=0 , which should be the time the ball was kicked up (or shot up).

b.) The function h=-16t%5E2%2B50t is a quadratic function that has a maximum for a certain value of t , and that value represents when the ball reaches its maximum height.
If you want to use a formula given in class, there is no need to think, you just apply the memorized formula.
Just in case, the formula said the the maximum for a function y=ax%5E2%2Bbx%2Bc with a%3C0 happens when
x=%28-b%29%2F%222+a%22 .
Of course, with h=-16t%5E2%2B50t ,
you have h instead of y and t instead of x ,
but it is the same kind of unction, with a=-16%3C0 , b=50 and c=0 .
So, the maximum occurs when t=%28-50%29%2F%282%2A%28-16%29%29=50%2F32=1.5625
The ball reaches maximum height after highlight%281.5625seconds%29 .
Hey, the means that when the ball meets the balloon (at t=1.875 seconds), the ball is already on its way down.
If you do not like to memorize stuff just because someone said so,
h=-16t%5E2%2B50t-->-h%2F16=t%5E2-%2850%2F16%29t-->-h%2F16=t%5E2-%2850%2F16%29t%2B%2850%2F32%29%5E2-%2850%2F32%29%5E2-->-h%2F16=%28t-50%2F32%29%5E2-%2850%2F32%29%5E2-->h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2 .
For t%3C%3E20%2F32 , -16%28t-50%2F32%29%5E2%3C0 and h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2%3C16%2850%2F32%29%5E2 .
For t=20%2F32 , -16%28t-50%2F32%29%5E2=0 and h=16%2850%2F32%29%5E2=2500%2A16%2F32%5E2=2500%2F64=39.0625 has its maximum value.


c.) The ball is at groune level when h=0 .
h=-16t%5E2%2B50t-->h=%28-16t%2B50%29t-->h=-16%28t-50%2F16%29t
So h=0 for system%28t=0%2C%22or%22%2Ct=16%2F50=3.125%29
So the ball leaves the ground at t=0 and hits the ground after highlight%283.125seconds%29 .

Here is a graph:


NOTE:
Math is use every day (by a few people) to do the calculations they need to do in real life. Some people think they do not need math.
Thirty-five years ago most people thought they did not need a microwave oven, because they did not know about them and they did not know how to use them. I spent a lot of money on one and enjoyed efficient cooking for the next 21 years. (Then it broke down and I got a new one). By then, most of those who said a microwave oven was not necessary had changed their mind. Would that ever happen with people who think you do not need math?