applications of quadratic function

Question

Tim is the punter for the bitterroot springs mustangs football team. he wrote the function h(t)=16t^2+8t+1 that he thinks will give the height of a football in terms of t, the number if seconds after he kicks the ball. use two different methods to determine the values of t for which h(t)=0 Is tim's function correct ?

I do not get what the t will give me 

Answer 1

Fiona, I hope I am not too late!

 

Let's say that f(x) = 2x - 5, then f(2) = 2(2) - 5 = -1.  

 

This problem is giving us the opposite.  It wants us to find t if h(t) = 0.  Since the function gives us the height of the ball, then the height being 0 means both when he kicked the ball and when it is caught.

 

16t² + 8t + 1 = 0.  We now need to solve for t.

 

One way we can do it is by factoring.  We multiply the first and third coefficient, 16 x 1 = 16.  We need to factors of 16 that give us 8 when added together.

 

Wouldn't ya know it, 4 x 4 = 16 and 4 + 4 = 8

 

So, we split 8t into 4t + 4t.  

 

16t²  + 4t + 4t + 1 = 0

 

Now, we factor out what is in common in the first two terms and the second two terms.

 

4t is common among the first two terms.  4t(4t + 1) = 16t² + 4t

 

Now, there is nothing in common in the second two terms besides 1, but we'll still put it out front.  1(4t + 1) = 4t + 1

 

Now, we combine.  4t(4t + 1) + 1(4t + 1) = 0

 

Since what is inside the two sets of parentheses is the same, we can combine the outside into its own set of parentheses, kind of like doing factoring backward.

 

(4t + 1)(4t + 1) = 0.

 

One or both terms on the left has to equal 0 for the right to be 0, right?  But they are both the same, so we only have to do it once.

 

4t + 1 = 0

      4t = -1

        t = -1/4

 

We can also use the quadratic formula to get the same answer.  I think you can do that by yourself.

 

So, h(-1/4) = 0.  This means that he kicks the ball and/or the ball is caught when the time is -1/4.  Unfortunately, we don't measure time in negative numbers, so his function is not correct!

 

Hope this helps!

 

 

Answer 2

David's explanation is great for the technical solution.  The question says use two different ways, so let me give you the other - substitution and common sense.  

 

t is the time, right?  so check where the ball is at t=0, t=1, t=2 and so on.  

 

At t=0, the ball is being kicked at 16(0^2) +8(0) + 1 = 1 foot off the ground.

 

At t=1 second, the ball is now 16(1^2) + 8(1) + 1 = 25 feet off the ground.

 

At t=2 seconds, the ball is now 16(2^2) + 8(2) + 1 = 32 + 16 + 1 = 49 feet off the ground.

 

At t=10 seconds, the ball is now 16(10^2) + 8(10) + 1 = 1600 + 80 + 1 = 1681 feet off the ground.

 

Since the coefficients, 16 and 8, are positive, every second the ball is rising.  That ball is never coming down.  

 

Tim has the wrong formula.

 

As a physics aside, that square term should be a -16, and the 8 in the t term is only right if Tim's kick gives the ball an initial upward speed of 8 feet per second - a really terrible kick.