Answer:
Step-by-step explanation:
The height of the tunnel is modeled by:
Where r and t are constants.
We know that the maximum height of the tunnel h is 6 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.
The vertex is given by:
In our case, we have the function:
Hence, a=r; b=t; and c=0.
Therefore, our vertex is:
Thus, if we substitute this back into our equation, we should get 6 since 6 is the maximum height which is determine by the vertex. In other words:
Simplify:
Simplify:
Combine fractions:
Multiply both sides by the denominator:
Let's solve for the constant r. Divide both sides by 24. Hence:
Now, we can use our knowledge of the width.
We know that the width is 12 meters at ground level.
Hence, when h=0, the <em>difference of our roots</em> is 12.
So:
We can factor:
By the Zero Product Property:
So, the first zero is 0.
Therefore, the second zero <em>must be 12</em> to ensure that our width is 12.
Let's isolate the second zero. Subtract t from both sides:
Divide both sides by r:
We know that this zero must be 12. Thus:
We have previously solved for r. Substitute:
Simplify:
Take the reciprocal of both sides:
Multiply both sides by 24. Hence, the value of t is:
Now, we can find r. r is given by:
By substituting 2 for t, then, we acquire that:
Therefore, for:
Our equation is: