Calculating Vehicle Height In A Semielliptical Tunnel A Mathematical Approach
#introduction In the realm of mathematics, practical problems often intertwine with geometrical concepts. This article delves into one such problem, exploring the application of semielliptical geometry in determining the maximum height of a vehicle that can safely pass through a tunnel. The scenario involves a tunnel with a semielliptical cross-section, a common structural design in engineering. By understanding the properties of ellipses and applying algebraic equations, we can calculate the height restriction at a specific point within the tunnel. This exercise not only reinforces our understanding of mathematical principles but also highlights their relevance in real-world applications, such as transportation and infrastructure development. Let's embark on this mathematical journey to unravel the solution.
Understanding the Problem
The problem presents a scenario where a road passes through a tunnel. This tunnel isn't just any ordinary passage; it has a unique shape: a semielliptical cross-section. This means that if you were to slice the tunnel perpendicular to the road, the resulting shape would be half of an ellipse. The tunnel's dimensions are crucial: it's 64 feet wide and 18 feet high at the center. The core question we aim to answer is: What is the height of the tallest vehicle that can pass through the tunnel at a point 22 feet from the center?
To tackle this, we need to understand the geometry of an ellipse and how its dimensions influence the space within it. The width of the tunnel corresponds to the major axis of the ellipse, and the height at the center corresponds to the semi-minor axis. The position 22 feet from the center is a horizontal distance along the major axis. Our task is to find the vertical height of the ellipse at this specific horizontal position. This involves using the equation of an ellipse and substituting the known values to solve for the unknown height. The solution will give us the maximum height a vehicle can have to pass safely at that point.
Setting up the Ellipse Equation
To solve this problem effectively, we first need to establish the equation that describes the semielliptical cross-section of the tunnel. An ellipse centered at the origin (0, 0) in a Cartesian coordinate system has a standard equation:
(x^2 / a^2) + (y^2 / b^2) = 1
Where:
- x and y are the coordinates of any point on the ellipse.
- a is the semi-major axis (half the width of the ellipse).
- b is the semi-minor axis (half the height of the ellipse).
In our scenario, the tunnel is 64 feet wide, which means the major axis is 64 feet. Therefore, the semi-major axis, a, is half of this width, which is 32 feet. The tunnel is 18 feet high at the center, which gives us the semi-minor axis, b, as 18 feet. Now we can plug these values into the equation:
(x^2 / 32^2) + (y^2 / 18^2) = 1
This equation mathematically represents the shape of our tunnel's cross-section. It allows us to relate the horizontal distance (x) from the center to the vertical height (y) at any point within the tunnel. This relationship is crucial for determining the maximum vehicle height at the specified distance from the center.
Solving for the Height
Now that we have the equation representing the semielliptical tunnel, we can proceed to calculate the height at a point 22 feet from the center. This means we will substitute x = 22 into our equation and solve for y, which will give us the height of the tunnel at that point. Our equation is:
(x^2 / 32^2) + (y^2 / 18^2) = 1
Plugging in x = 22, we get:
(22^2 / 32^2) + (y^2 / 18^2) = 1
First, let's simplify the equation by calculating the squares:
(484 / 1024) + (y^2 / 324) = 1
Now, we want to isolate the term with y, so we subtract (484 / 1024) from both sides:
(y^2 / 324) = 1 - (484 / 1024)
To subtract the fractions, we need a common denominator. In this case, it's 1024:
(y^2 / 324) = (1024 / 1024) - (484 / 1024)
(y^2 / 324) = 540 / 1024
Next, we multiply both sides by 324 to isolate y^2:
y^2 = (540 / 1024) * 324
y^2 = 174960 / 1024
y^2 ≈ 170.86
Finally, we take the square root of both sides to solve for y:
y ≈ √170.86
y ≈ 13.07
So, the height of the tunnel at a point 22 feet from the center is approximately 13.07 feet. This means that the tallest vehicle that can pass through the tunnel at this point should be slightly less than 13.07 feet to ensure safe passage.
Selecting the Correct Option
After calculating the height of the tunnel at a point 22 feet from the center, we found that the maximum height for a vehicle to pass safely is approximately 13.07 feet. Now, let's revisit the options provided:
A. 16 ft B. 13 ft C. 21 ft D. 18 ft
Comparing our calculated height of 13.07 feet to the options, we can see that option B, 13 ft, is the closest and most reasonable answer. While our calculated value is slightly higher, the options are given in whole numbers, and 13 feet is the nearest whole number that is less than our calculated height. This ensures that a vehicle with this height would safely pass through the tunnel at the specified point.
Therefore, the correct answer is B. 13 ft. This selection aligns with our calculated result and the practical constraints of the problem.
Practical Implications and Further Considerations
Our solution to this problem has significant practical implications, especially in the fields of civil engineering and transportation planning. Understanding the maximum vehicle height that can safely pass through a tunnel at a given point is crucial for designing roads and infrastructure that can accommodate various types of vehicles. It ensures the safe and efficient movement of traffic, preventing accidents and damage to both vehicles and the tunnel structure.
Beyond the immediate answer, this problem also highlights several factors that engineers and planners must consider in real-world scenarios. These include:
- Safety Margins: In practice, engineers often incorporate safety margins to account for uncertainties and variations. For instance, they might specify a maximum vehicle height slightly lower than the calculated value to provide an additional buffer.
- Vehicle Dimensions: Different types of vehicles have varying heights. Trucks, buses, and recreational vehicles (RVs) are typically taller than passenger cars. Planners must consider the mix of vehicles that will use the tunnel and design accordingly.
- Tunnel Design: The shape and dimensions of the tunnel itself can vary. While we've considered a semielliptical cross-section, other shapes are possible. The design must balance structural integrity, cost-effectiveness, and the need to accommodate traffic.
- Roadway Alignment: The alignment of the road within the tunnel can also affect vehicle clearance. Curves and changes in elevation can reduce the available headroom, requiring adjustments to the tunnel's design.
- Maintenance and Wear: Over time, tunnels can undergo deformation or wear, which may affect their dimensions. Regular inspections and maintenance are essential to ensure that clearances remain within safe limits.
Conclusion
In conclusion, we have successfully determined the height of the tallest vehicle that can pass through a tunnel with a semielliptical cross-section at a specific point from the center. By applying the equation of an ellipse and performing algebraic calculations, we found that a vehicle with a height of approximately 13 feet can safely pass through the tunnel at a point 22 feet from the center. This exercise demonstrates the practical application of mathematical concepts in real-world engineering problems.
Furthermore, our discussion has highlighted the importance of considering various factors in tunnel design and transportation planning. Safety margins, vehicle dimensions, tunnel geometry, roadway alignment, and maintenance are all crucial aspects that engineers and planners must address to ensure the safe and efficient use of infrastructure. By integrating mathematical principles with practical considerations, we can create solutions that meet the needs of society while upholding safety and sustainability standards.
