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.

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TopNewestDo you mean the red lines indicated? Or there are some other blue lines that is not shown? – 敬全 钟 · 3 years, 8 months ago

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– Oussama Jaber · 3 years, 8 months ago

Sorry I meant the read line.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. – Daniel Liu · 3 years, 8 months ago

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have four lanes semi trucks go on the 2 enter lanes even tho its not safe – Curtis Noble · 1 year, 11 months ago

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is it one way or two day road ? – Anq Anq · 3 years, 8 months ago

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– Oussama Jaber · 3 years, 8 months ago

one wayLog 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. – Samarth M.O. · 3 years, 8 months ago

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\(=\quad \{ (l,\quad h):\quad \frac { { l }^{ 2 } }{ 4 } +{ h }^{ 2 }\le { R }^{ 2 },\quad 2\le l\le 2.40,\quad 3.10\le h\le 3.30\} \) – Lokesh Sharma · 3 years, 8 months ago

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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 . – Mohamed Mohamed Dehairy · 3 years, 8 months ago

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Is the height of the truck tires significant? – Daniel Liu · 3 years, 8 months ago

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– Oussama Jaber · 3 years, 8 months ago

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? – Oussama Jaber · 3 years, 8 months ago

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– Lokesh Sharma · 3 years, 8 months ago

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

– Oussama Jaber · 3 years, 8 months ago

I meant red lineLog in to reply