Question

Parabola Equations - Algebra

Please explain working clearly.
Help greatly appreciated,
Thanks!

Answer 1

`\displaystyle h=-(1)/(6)x^2+2x`

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(-(b)/(2a), f\left(-(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 = -(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(-(t)/(2r)\right)^2+t\left(-(t)/(2r)\right)`

Simplify:

`\displaystyle 6=r\left((t^2)/(4r^2)\right)-(t^2)/(2r)`

Simplify:

`\displaystyle 6=(t^2)/(4r)-(t^2)/(2r)`

Subtract:

`\displaystyle 6=(t^2)/(4r)-(2t^2)/(4r) = -(t^2)/(4r)`

Multiply:

`\displaystyle 24r=-t^2, \, r\\eq 0`

Solve for r:

`\displaystyle r=-(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=-(t)/(r)`

We know that this zero must be 12. Thus:

`\displaystyle 12=-(t)/(r)`

We have previously solved for r. Substitute:

`\displaystyle 12=-(t)/(-(t^2)/(24))`

Simplify:

`\displaystyle 12=(24)/(t)`

Take the reciprocal of both sides:

`\displaystyle (t)/(24)=(1)/(12)`

Multiply. Hence, the value of t is:

`t=2`

Find r. Recall that:

`\displaystyle r=-(t^2)/(24)`

Therefore:

`\displaystyle r=-((2)^2)/(24)=-(4)/(24)=-(1)/(6)`

Then in conclusion, our equation will be:

`\displaystyle h=-(1)/(6)x^2+2x`