Problem of the Day

The height of a model rocket in feet, $x$ , $t$ seconds after its launch is modeled by the equation $x=-t^2+bt+a$ , where $a$ and $b$ are positive integers. At what value of $t$ does the rocket reach its maximum height?

(1) $b=10$

(2) $a=125$

Reveal Answer

Answer

A. Statement 1 alone is sufficient, but statement 2 alone is not sufficient.

See the Solution

Solution

[latexpage]

Statement 1: SUFFICIENT. $b=10$ tells us that the equation is $x=-t^2+10t+a$. We can rewrite this by completing the square as follows:

$-t^2+10t+a$

$-(t^2-10t)+a$

$-(t^2-10t+25)+a+25$

$-(t-5)^2+a+25$

So, if $x=-(t-5)^2+a+25$, $x$ will be maximized when $t=5$. Hence, statement 1 is sufficient.

Statement 2: INSUFFICIENT. The value of $a$ will have some impact on how high the rocket goes, but it will have no impact on what time it will reach its maximum height. To find that, we have to have a value of $b$ so that we can complete the square as we did for statement 1.

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