We are tasked with finding the maximum height a vehicle, 6 feet wide, can have in order to safely drive under a semi-elliptical tunnel that is 27 feet high and 33 feet wide.

Step 1: Set up the equation of the ellipse

The equation of an ellipse in standard form is given by:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

where:

• $a$ is half the width of the tunnel (semi-major axis),
• $b$ is the height of the tunnel (semi-minor axis),
• $x$ and $y$ are coordinates along the width and height of the ellipse, respectively.

Given:

• Full width of the tunnel = 33 feet, so $a = \frac{33}{2} = 16.5$ feet,
• Full height of the tunnel = 27 feet, so $b = 27$ feet.

Thus, the equation of the tunnel's semi-elliptical shape is:

$\frac{x^2}{(16.5)^2} + \frac{y^2}{(27)^2} = 1$

or:

$\frac{x^2}{272.25} + \frac{y^2}{729} = 1$

Step 2: Determine the height at $x = 3$ feet

The vehicle is 6 feet wide, which means it extends 3 feet to the left and right of the center of the tunnel. We need to find the height of the tunnel at $x = 3$ feet.

Substitute $x = 3$ into the equation of the ellipse:

$\frac{3^2}{272.25} + \frac{y^2}{729} = 1$

$\frac{9}{272.25} + \frac{y^2}{729} = 1$

$0.03305 + \frac{y^2}{729} = 1$

$\frac{y^2}{729} = 1 - 0.03305 = 0.96695$

$y^2 = 0.96695 \times 729 = 704.076$

$y = \sqrt{704.076} \approx 26.54$

Step 3: Conclusion

The height of the tunnel at $x = 3$ feet (3 feet from the center) is approximately 26.54 feet. Therefore, the maximum height of the vehicle must be less than or equal to 26.54 feet to safely pass through the tunnel.

Follow-up Questions:

1. How does the shape of an ellipse affect the height at various points?
2. What if the width of the vehicle increased to 8 feet? How would this change the maximum allowable height?
3. How would the answer change if the tunnel had a different height, say 30 feet?
4. Can this approach be generalized to tunnels of other shapes, such as parabolic?
5. What would happen if the tunnel's width was narrower than 33 feet but the height remained 27 feet?

Tip: Always ensure that units are consistent when performing calculations. It’s important to double-check the shape's dimensions before starting any work!