Question

Parabola Equations - Algebra
Please explain working clearly.
Help greatly appreciated,
Thanks!

Answer 1

Answer:

$\displaystyle h=-\frac{1}{6}x^2+2x$

Step-by-step explanation:

We are given that the height of the tunnel is modeled by:

$h=rx^2+tx$

Where r and t are constants.

We are also given that the maximum height of the tunnel h is six meters.

And at ground level, the width is 12 meters.

And we want to determine the equation of the parabola.

First, since this is a quadratic, our maximum height h will occur at the vertex of our equation.

Recall that the vertex is given by the formulas:

$\displaystyle \text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

In our case, we have the function:

$h=(r)x^2+(t)x$

Hence, a = r; b = t; and c = 0.

Therefore, the x-coordinate of our vertex is:

$\displaystyle x = -\frac{t}{2r}$

Therefore, if we substitute this back into our equation, the result should be six since six is the maximum height. Hence:

$\displaystyle (6)=r\left(-\frac{t}{2r}\right)^2+t\left(-\frac{t}{2r}\right)$

Simplify:

$\displaystyle 6=r\left(\frac{t^2}{4r^2}\right)-\frac{t^2}{2r}$

Simplify:

$\displaystyle 6=\frac{t^2}{4r}-\frac{t^2}{2r}$

Subtract:

$\displaystyle 6=\frac{t^2}{4r}-\frac{2t^2}{4r} = -\frac{t^2}{4r}$

Multiply:

$\displaystyle 24r=-t^2, \, r\neq 0$

Solve for r:

$\displaystyle r=-\frac{t^2}{24}$

Next, we are given that the width is 12 meters at ground level.

Hence, when h = 0, the difference of our roots is 12:

We have:

$0=rx^2+tx$

Factor:

$0=x(rx+t)$

By the Zero Product Property:

$x=0\text{ or } rx+t=0$

Hence, the first zero is 0.

Therefore, the second zero must be 12 to ensure that our width is 12.

Let's isolate the second zero. Subtract t from both sides:

$rx=-t$

Divide both sides by r:

$\displaystyle x=-\frac{t}{r}$

We know that this zero must be 12. Thus:

$\displaystyle 12=-\frac{t}{r}$

We have previously solved for r. Substitute:

$\displaystyle 12=-\frac{t}{-\dfrac{t^2}{24}}$

Simplify:

$\displaystyle 12=\frac{24}{t}$

Take the reciprocal of both sides:

$\displaystyle \frac{t}{24}=\frac{1}{12}$

Multiply. Hence, the value of t is:

$t=2$

Find r. Recall that:

$\displaystyle r=-\frac{t^2}{24}$

Therefore:

$\displaystyle r=-\frac{(2)^2}{24}=-\frac{4}{24}=-\frac{1}{6}$

Then in conclusion, our equation will be:

$\displaystyle h=-\frac{1}{6}x^2+2x$