a semicircular tunnel covers 2 cemented lanes. beyond the lanes are the gutters whose widths of 1ft. each extend up to each of the edges of the tunnel. the maximum height of vehicles that can pass through the tunnel while traveling along the edge of the lane is 4sqrt2 ft. what is the total with(ft.) of the tunnel?

To solve this problem, let’s break it down step by step:

### Step 1: Represent the Semicircular Tunnel
The tunnel is semicircular, so we know that its cross-section resembles a half-circle.

Let:
- \( r \) be the radius of the tunnel.
- The total width of the tunnel (diameter) would be \( 2r \).

### Step 2: Analyze the Given Information
1. **The total width of the lanes and gutters:**
   - The problem states that the tunnel covers **2 lanes** and **2 gutters**. 
   - Each **lane** is cemented and extends between the gutters.
   - The **gutters** are 1 foot wide each, and the lanes lie between the gutters.

Let's define:
   - \( w \) as the width of one lane.
   - The total width of the lanes and gutters is:
     \[
     \text{Total width} = 2 \times 1 \text{ (gutter width)} + 2 \times w \text{ (lane width)} = 2 + 2w.
     \]

2. **Vehicle height restriction:**
   - The maximum height of vehicles traveling along the edge of the lane (near the gutters) is given as \( 4\sqrt{2} \) feet.
   - This means that the vertical distance from the lane’s edge to the top of the tunnel is \( 4\sqrt{2} \) feet.

### Step 3: Geometry of the Problem
Since the tunnel is semicircular, the relationship between the height of the vehicle and the radius of the tunnel can be analyzed as follows:
- At a distance of \( w + 1 \) feet (from the center of the tunnel to the edge of the lane plus gutter), the height is \( 4\sqrt{2} \) feet.

Using the Pythagorean theorem in the right triangle formed by the radius \( r \), the distance from the center to the edge \( w + 1 \), and the height \( 4\sqrt{2} \), we have:
\[
r^2 = (w + 1)^2 + (4\sqrt{2})^2.
\]
Simplifying the height term:
\[
(4\sqrt{2})^2 = 16 \times 2 = 32,
\]
so the equation becomes:
\[
r^2 = (w + 1)^2 + 32.
\]

### Step 4: Solving for \( w \) and \( r \)
We still need to find more information to solve for the width \( w \). However, given the setup of the problem, it's evident that more assumptions or data on the lane width or radius might be needed to compute the total width of the tunnel.

---

Would you like further details, or do you have any follow-up questions?

### Relative Questions:
1. How can we use the Pythagorean theorem in more complex tunnel shapes?
2. If the lane width was known, how would we calculate the total tunnel width directly?
3. What happens if the tunnel was elliptical instead of semicircular?
4. How does vehicle height affect tunnel design for different types of vehicles?
5. What role do gutters play in the overall tunnel width calculation?

### Tip:
When dealing with semicircular structures, the Pythagorean theorem is often helpful for connecting distances along the radius and height within the structure.

A semicircular tunnel covers 2 cemented lanes. Beyond the lanes are the gutters whose widths of 1ft. each extend up to each of the edges of the tunnel. The maximum height of vehicles that can pass through the tunnel while traveling along the edge of the lane is 4√2 ft. What is the total width (ft.) of the tunnel?

This problem involves calculating the total width of a semicircular tunnel that covers two lanes and gutters, given that the maximum height of vehicles traveling along the lane edge is 4√2 feet. Learn how to use the Pythagorean theorem to find the relationship between tunnel height and width. Suitable for Grades 10-12 geometry students.

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