## Task B-2.7.

The cross-section of the tunnel has the shape of a parabola. The maximum width of the tunnel is $6 \mathrm{~m}$, and the maximum height is $8 \mathrm{~m}$. Can a truck that is $4 \mathrm{~m}$ wide and $4.5 \mathrm{~m}$ tall pass through this tunnel? Justify your answer! If the height of the truck is $4 \mathrm{~m}$, what is the maximum width it can have to pass through the tunnel?
## Solution.

We place the parabola in the coordinate system so that its vertex has coordinates $(0,8)$.

![](https://cdn.mathpix.com/cropped/2024_05_30_c3f5968a3447b3c0b4b2g-09.jpg?height=607&width=659&top_left_y=713&top_left_x=706)

Then its roots are $T_{1}(-3,0)$ and $T_{2}(3,0)$, and the equation is

$$
y=-\frac{8}{9} x^{2}+8
$$

We can also accept other equations of the parabola (depending on the choice of the coordinate system) if it is clear how the student arrived at it, for example, $y=-\frac{8}{9} x^{2}+\frac{16}{3} x$.

If the width of the truck is 4 meters, then its height should be less than

$$
-\frac{8}{9} \cdot 2^{2}+8=\frac{40}{9}
$$

Since $4.5 > \frac{40}{9}$, the truck will not be able to enter the tunnel.

If the height of the truck is 4 meters, then the maximum width $(2 x)$ can be calculated from:

$$
\begin{aligned}
4 & < -\frac{8}{9} x^{2} + 8 \\
x^{2} & < \frac{9}{2}
\end{aligned}
$$

The maximum width must be less than $2 \sqrt{\frac{9}{2}} = 3 \sqrt{2}$.