Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The height of an object is given by $$ h(x) = -0.005x^2 + x. $$ When does the object hit the ground? When does it attain its maximum height? What is its maximum height?

I divided -.005/-1 to get 100, replaced x with 100, and ended up getting y = 50 when it hits the ground. That is what I got as the Range, when x = 0 (stop me if I'm wrong), or does -b/2a get me the maximum of the vertex? I'm, confusing myself now. If someone could give me a little explanation I would greatly appreciate it.

share|improve this question
    
How do you get 100 when you divide -.005 by -1? –  Gerry Myerson Mar 14 '12 at 6:12
    
@AustinMohr Yes to your question, sorry I was not clear. –  stytown Mar 14 '12 at 6:20

1 Answer 1

up vote 1 down vote accepted

From your equation $h(x)=-0.005x^2+x$, we get that $h(x)=0$ when $x(-0.005x+1)=0$. This happens when $x=0$, and also when $-0.005x+1=0$. Rewrite this as $0.005x=1$. We get $x=\frac{1}{0.005}=200$.

The maximum occurs halfway between $x=0$ and $x=200$, that is, at $x=100$. Substitution shows that $h(100)=50$.

Another way of doing it is by using a remembered formula. And indeed if $a=-0.005$ and $b=1$, the maximum height happens when $x=\frac{-b}{2a}$, which in this case gives $x=100$.

Then you substitute in the height formula to get the maximum height. The $50$ that you get is the maximum height, and has nothing to do with hitting the ground. For that, we got $x=200$.

Remark: I will add some theory that may help in the long run. We have $h(x)=-0.005x^2+x$. Note that $0.005=\frac{1}{200}$. So we can rewrite the formula for $h(x)$ as $$h(x)=-\frac{1}{200}(x^2-200x).$$ Complete the square. We get $$h(x)=-\frac{1}{200}((x-100)^2 -10000)$$ or more simply $$h(x)=50-\frac{1}{200}(x-100)^2.$$ The term $\frac{1}{200}(x-100)^2$ is always $\ge 0$, and is $0$ when $x=100$. At that value of $x$, $h(x)$ is as big as it will ever get, namely $50$.

share|improve this answer
    
Thanks, very clear explanation, exactly what I was looking for. –  stytown Mar 14 '12 at 6:21
    
Double thanks for the even broader explanation! –  stytown Mar 14 '12 at 6:29

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.