SOLUTION: I do not know where to start. A red ball and a green ball are simultaneously tossed into the air. Find a polynomial D(t) that represents the difference in the heights of the two

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Question 262846: I do not know where to start.
A red ball and a green ball are simultaneously tossed into the air.
Find a polynomial D(t) that represents the difference in the heights of the two balls. The red ball is given an initial velocity of 96 feet per second, and its height t seconds after it is tossed is -16t^2 + 96t feet. The green ball is given an initial velocity of 80feet per second, and its height t second, and its height t seconds after it is tossed is -16t^ + 80t feet. How much higher is the red ball 2 seconds after the balls are tossed? In reality, when does the difference in the heights stop increasing?
Found 2 solutions by ankor@dixie-net.com, drk:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Find a polynomial D(t) that represents the difference in the heights of the two balls.
The red ball is given an initial velocity of 96 feet per second, and its height t seconds after it is tossed is -16t^2 + 96t feet.
The green ball is given an initial velocity of 80feet per second, and its height t second, and its height t seconds after it is tossed is -16t^ + 80t feet.
How much higher is the red ball 2 seconds after the balls are tossed?
In reality, when does the difference in the heights stop increasing?
:
D(t) = (-16t^2 + 96t) - (-16t^2 +80t)
:
Remove brackets
D(t) = -16t^2 + 96t + 16t^2 - 80t
:
combine like terms:
D(t) = 16t
:
How much higher is the red ball 2 seconds after the balls are tossed?
D(t) = 16(2)
D(t) = 32 ft difference after 2 seconds
:
When does the difference in the heights stop increasing?
:
Obviously, when one of the balls hits the ground
:
Find when the lowest ball hits the ground (h=0)
-16t^2 + 80t = 0
Factor out -16t
-16t(t - 5) = 0
t = 5 seconds when the difference stops increasing;
:
A graph illustrates this well
+graph%28+300%2C+200%2C+-4%2C+8%2C+-20%2C+150%2C+-16x%5E2%2B96x%2C+-16x%5E2%2B80x%29+

Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
Lets look at the red ball equation as
(i) h%28t%29+=+-16t%5E2+%2B+96t
we want to express this in vertex form to get the max ht. in vertex form the red ball is
(ii) h%28t%29+=+-16%28t-3%29%5E2+%2B+144
this tells us that (1) it is a parabola opening down; (2) after 3 seconds a max ht occurs at 144 feet.
t= 2 gives us
h%282%29+=+-16%282-3%29%5E2+%2B+144
h%282%29+=+128+ft.
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Lets look at the green ball equation as
(i) h%28t%29+=+-16t%5E2+%2B+80t
we want to express this in vertex form to get the max ht. in vertex form the green ball is
(ii) h%28t%29+=+-16%28t-2.5%29%5E2+%2B+100
this tells us that (1) it is a parabola opening down; (2) after 2.5 seconds a max ht occurs at 100 feet.
t= 2 gives us
h%282%29+=+-16%282-2.5%29%5E2+%2B+100
h%282%29+=+96+ft.
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Since the red ball had a great initial velocity, we have the difference formula as
D(t) = red ball equation - green ball equation
or
D(t) = -16(t-3)^2 + 144 - (-16(t-2.5)^2 + 100))
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after 2 seconds, the red ball is 32 feet higher than the green ball.
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the difference in heights will always be increasing