a tunnel has the shape of a semi ellipse that is 15 ft high at the center, and 36 ft across at the base. at most, how high should a passing truck be if it is 12 ft wide, for it to fit through the tunnel?

To solve this problem, we need to model the shape of the tunnel and the truck's position as a mathematical function.

The tunnel is in the shape of a semi-ellipse. The equation of an ellipse with its center at the origin is:

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

where:
- \(a\) is the horizontal semi-axis (half of the base),
- \(b\) is the vertical semi-axis (the height at the center).

Given:
- The width of the base of the tunnel is 36 feet, so \(a = \frac{36}{2} = 18\) feet.
- The height at the center of the tunnel is 15 feet, so \(b = 15\) feet.

Thus, the equation of the semi-ellipse representing the tunnel is:

\[
\frac{x^2}{18^2} + \frac{y^2}{15^2} = 1
\]

We need to find the maximum height a truck can have if its width is 12 feet. Since the truck is centered on the x-axis, its sides are 6 feet from the center, so we need to find the height of the tunnel at \(x = 6\) feet.

Substitute \(x = 6\) into the equation of the ellipse:

\[
\frac{6^2}{18^2} + \frac{y^2}{15^2} = 1
\]

\[
\frac{36}{324} + \frac{y^2}{225} = 1
\]

Simplifying:

\[
\frac{1}{9} + \frac{y^2}{225} = 1
\]

\[
\frac{y^2}{225} = 1 - \frac{1}{9} = \frac{8}{9}
\]

\[
y^2 = 225 \times \frac{8}{9} = 200
\]

\[
y = \sqrt{200} \approx 14.14
\]

Therefore, the maximum height the truck can be to pass through the tunnel is approximately 14.14 feet.

### Do you need more details, or have any questions?

Here are some related questions:
1. How would the answer change if the tunnel were wider or narrower?
2. What happens if the truck is not centered within the tunnel?
3. How would the calculation differ if the tunnel had a different geometric shape?
4. Can the equation of a semi-ellipse be derived from the properties of a full ellipse?
5. What if the truck had a height limit imposed by other factors, such as regulations?

**Tip:** Always confirm units and dimensions are consistent before starting calculations.

A tunnel has the shape of a semi ellipse that is 15 ft high at the center, and 36 ft across at the base. At most, how high should a passing truck be if it is 12 ft wide, for it to fit through the tunnel?

Discover how to calculate the maximum height of a truck that can pass through a semi-elliptical tunnel with a height of 15 feet at the center and a base width of 36 feet. This problem covers geometric concepts related to ellipses and includes a detailed step-by-step solution. Ideal for high school students in Grades 10-12.

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