Lena is in a cave whose floor measures $20\times20$ m, and is $30$ m tall. She wants to determine an equation that describes the path of an object in the air. Her experiment consists of launching a measuring device from the midpoint of one side of the cave to the midpoint of the opposite side of the cave. The device starts on the floor, and lands on the floor. The device takes the following readings: $(25,2)$, $(6,32)$, and $(3,12)$. The first coordinate is the distance from midpoint to midpoint, in meters, and the second coordinate is the height, in meters. But the device is faulty, and only one of the readings is correct. After some thought, Lena is able to determine her equation.
a) Which of the readings is correct? Explain in full sentences, using appropriate key words.
b) What is her final equation? Show your work.
For question a), I know that $(3,12)$ is the only right coordinate because the room measures $20$ by $30$ and the other two coordinates surpass $20$ (as width) and $30$ (as height).
I'm confused on part b). Here's the first way I think you could do it:
I was thinking that since the horizontal distance of the room is $20$, I assumed that $0$ are $20$ the $x$-intercepts of the parabola and plugged those into the formula $y=a(x-r)(x-s)$, plugging in $0$ for $r$ and $20$ for $s$, yielding $y=ax(x-20)$. Then I plugged in $3$ and $12$ for $x$ and $y$, respectively. That gives $12=a(3)(3-12)$, and solving for $a$, I got $a=-4/17$. I then plugged $a=-4/17$ into $y=ax(x-20)$, and I got $y=-\frac{4}{17}x^2+4.7$. I know the $x$-coordinate of the vertex of my parabola is $10$ because $(0+20)/2=10$. So the final equation Lena determines is $y=-\frac{4}{17}x^2+4.7$, with the vertex being $(10,23.5)$.
I'm not sure if this is the correct way to do it, so there is another way you could possibly do it below.
If you assume $30$ is the maximum height of the parabola and you plug it into the equation $y=a(x-h)^2+k$, and you plug $10$ in for $h$ (because the $x$-coordinate of the vertex is $10$), while also plugging in $3$ and $12$ for $x$ and $y$, respectively, you get $12=a(3-10)^2+30$. Then you solve for $a$ and plug it into $y=a(x-10)^2+30$, and that would be your final equation (with a vertex of $(10,30)$.
Both ways have different final equations, and I need help figuring out which one is the right way or if there is another way that will get me the right equation. Thanks!
Your first way is definitely the right way to approach the problem. We can't assume that the measuring device actually reaches the top of the cave (i.e. that the $y$-coordinate of the vertex of the parabola is $30$), so building an equation based on that won't work. It's explicitly stated in the problem, however, that the device does go from one end of the cave to the other, so it's perfectly fair to assume that the $x$-intercepts of the parabola are $0$ and $20$.
As for the equation you reach, that's not quite right. Your reasoning is fine, but I think something went wrong in your calculations. First of all, there's a typo in your work; you should solve the equation $12=a(3)(3-20)$ for $a$, not $12=a(3)(3-12)$. But seeing as you got $a=-4/17$, you did solve the correct equation, even if you wrote the wrong one. But secondly, and more importantly, something must have gone wrong in your simplification after plugging $a$ back in. $y=-\frac{4}{17}x^2+4.7$ is not equal to $y=-\frac{4}{17}x(x-20)$, which is what you get after plugging $a$ back in and without simplifying at all.
But $y=-\frac{4}{17}x(x-20)$ is the equation you're looking for, and we can check by seeing if it fits all the necessary criteria. Its $x$-intercepts are $(0,0)$ and $(20,0)$, and it passes through the point $(3,12)$, so it's all good.
(Side note: There's only ever one parabola passing through any three points, so the simple fact that all three of those points are on our parabola is proof enough that it's the right parabola.)