SOLUTION: The equation y = -16t^2 - 18t + 405 describes the height (in feet) of a ball thrown downward at 18 feet per second from a height of 405 feet from the ground. In how many seconds wi

Question 1090463: The equation y = -16t^2 - 18t + 405 describes the height (in feet) of a ball thrown downward at 18 feet per second from a height of 405 feet from the ground. In how many seconds will the ball hit the ground? Express your answer as a decimal rounded to the nearest tenth.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

When the ball hits the ground, its height is 0. So you want to solve the equation

The question asks for an answer to the nearest tenth, suggesting that the answer is not a "nice" number and has to be rounded. However, the quadratic expression factors, yielding an exact answer.

You could just use the quadratic formula, or perhaps a graphing calculator or some other such tool, to find the answer. But let's instead get some practice with this problem on factoring quadratics.

I would first factor out a -1, so that the leading coefficient is positive; factoring a quadratic with a negative leading coefficient I find to be a very unpleasant task. So our quadratic is

With a leading coefficient of 16, the leading coefficients of our two binomial factors could be 1 and 16, or 2 and 8, or 4 and 4.

But in fact they can't be 4 and 4 -- because then the coefficient of the linear (x) term would have to be a multiple of 4, which it is not.

And leading coefficients of 1 and 16 on the binomial factors is unlikely, so I would look first for a factorization with 2 and 8 as the leading coefficients of the two binomial factors.

Some of the possible factorizations of the constant term 405 are 81*5, 27*15, 9*45, .... Some playing around with the different possible combinations shows the factorization to be

Setting the first factor equal to 0 gives us a negative value for the time t, which does not make sense in the actual problem. Setting the second factor equal to 0 gives us the value of 4.5 for t.

So that is the answer to the problem: the ball hits the ground after 4.5 seconds.

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