Framing on R.....Can my truck pass through this tunnel?

A tunnel in the form of a semi cylinder has a radius of 4 m. A factory has several trucks whose widths L vary between 2m and 2.40m and whose heights vary between 3.10m and 3.30m.
Can all the trucks pass in this tunnel without exceeding the red line indicated in the figure?
The heights mentioned is from ground to the top of the trucks.

This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .

Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.

Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold

- bulleted - list

bulleted

list

1. numbered 2. list

numbered

list

Note: you must add a full line of space before and after lists for them to show up correctly

The truck with the largest cross-section diagonal is the one with the greatest dimensions; namely, 2.4m x 3.3m.
We have that $\sqrt{2.4^2+3.3^2}\approx 4.08$; but since the semi-cylinder can only fit up to 4 meters diagonally, not all the trucks can fit.

I am assuming that the truck must occupy one half of the cylinder or less, therefore following driving laws.

i assumed that the truck shouldn't exceed the line halving the tunnel width ,or the tunnel radius .
in this case if the maximum height is 3.3 m ,then the maximum width should be 3.3 m also ,because the cross section of the tunnel is a half circle ,so yes all the trucks can pass the tunnel .

The height of the truck mentioned includes the height of the truck. Sorry the picture should be modified so the height line should go all the way to the ground.

Distance from the center to the circumference stays the same. So, (l/2)^2 + h^2 <= r^2 where r is the radius.
I can't figure out the blue line in your picture.

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewesthave four lanes semi trucks go on the 2 enter lanes even tho its not safe

Log in to reply

is it one way or two day road ?

Log in to reply

one way

Log in to reply

ALL trucks can pass through the tunnel .Any truck with width 2.4 and height not greater than 3.815... can pass through the tunnel.

Log in to reply

$=\quad \{ (l,\quad h):\quad \frac { { l }^{ 2 } }{ 4 } +{ h }^{ 2 }\le { R }^{ 2 },\quad 2\le l\le 2.40,\quad 3.10\le h\le 3.30\}$

Log in to reply

The truck with the largest cross-section diagonal is the one with the greatest dimensions; namely, 2.4m x 3.3m. We have that $\sqrt{2.4^2+3.3^2}\approx 4.08$; but since the semi-cylinder can only fit up to 4 meters diagonally, not all the trucks can fit.

I am assuming that the truck must occupy one half of the cylinder or less, therefore following driving laws.

Log in to reply

i assumed that the truck shouldn't exceed the line halving the tunnel width ,or the tunnel radius . in this case if the maximum height is 3.3 m ,then the maximum width should be 3.3 m also ,because the cross section of the tunnel is a half circle ,so yes all the trucks can pass the tunnel .

Log in to reply

Do you mean the red lines indicated? Or there are some other blue lines that is not shown?

Log in to reply

Sorry I meant the read line.

Log in to reply

Is the height of the truck tires significant?

Log in to reply

The height of the truck mentioned includes the height of the truck. Sorry the picture should be modified so the height line should go all the way to the ground.

Log in to reply

All I could do is frame both L and h

$2 \leq L \leq 2.40$

$3.10 \leq h \leq 3.30$

What should I do next?

Log in to reply

Distance from the center to the circumference stays the same. So, (l/2)^2 + h^2 <= r^2 where r is the radius. I can't figure out the blue line in your picture.

Log in to reply

I meant red line

Log in to reply