SOLUTION: The height, h, of a cricket ball above the ground after t seconds is given by the eqaution h = 20t - 5t^2. When is the ball 25 metres above the ground? Need help! Don't know

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: The height, h, of a cricket ball above the ground after t seconds is given by the eqaution h = 20t - 5t^2. When is the ball 25 metres above the ground? Need help! Don't know       Log On


   

Question 241545: The height, h, of a cricket ball above the ground after t seconds is given by the eqaution h = 20t - 5t^2.
When is the ball 25 metres above the ground?

Need help! Don't know what to do! Maths exam tomorrow
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
So you are given the equation and the height and you've been asked to find the time. You just put the height into the equation and then use Algebra to solve for the time:
20+=+20t+-+5t%5E2
To solve this we need to recognize that, because of the t%5E2, it is a quadratic equation.

To solve a quadratic equation you get one side equal to zero and then factor it of use the Quadratic formula. So we will start by subtracting 20 from each side (the quickest way to get one side to be zero):
0+=+-5t%5E2+%2B+20t+-+20
(I've also rearranged the terms to make the factoring (or use of the Quadratic formula easier.) When factoring, always start by factoring out the Greates COmmon Factor (GCF) (unless it is a 1). The GCF here is 5:
0+=+5%28-t%5E2+%2B+4t+-+4%29
We could factor the trinomial factor as it is but I like having a positive leading coefficient so I will factor out a -1. (I could have factored out a -5 earlier but I didn't want to confuse you.)
0+=+5%28-1%29%28t%5E2+-4t+%2B+4%29
The trinomial is now more recognizably factorable. It fits the pattern: a%5E2+-2ab+%2B+b%5E2+=+%28a-b%29%28a-b%29+=+%28a-b%29%5E2. So it factors into:
0+=+5%28-1%29%28t+-+2%29%5E2
According to the Zero Product Property this product can be zero only if one of the factors is zero. 5 can't be zero and neither can -1. But t-2 can:
t-2 = 0
t = 2

So the cricket ball is 25 meters (metres in the UK) high after 2 seconds. (If you think about it, this must be the maximum height of the cricket ball. The cricket ball will be at the various heights once on the way up and once on the way down. So there should be two times for each height. The exception is the maximum height. The maximum height is achieved only once, in the middle of the flight. This is why we only got one time for this height.)