Log Length in L-shaped River Channel

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In summary, two rivers of widths 64m and 125m meet at a right angle, forming an L-shaped channel. The task is to determine the longest log that can be floated in the channel. The attempt at a solution involves drawing a picture and finding a formula for the length of a line in the water that passes through a given point. The goal is to find the minimum value of x, which can possibly be obtained using derivatives. It is suggested that this problem may require more than precalculus knowledge.
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GPhab
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Homework Statement



Two rivers of unequal width (64m and 125m) meet at a right angle, forming an L-shaped channel. What is the longest possible log you can float on it?

Homework Equations





The Attempt at a Solution


I tried for equal widths and built a recursive algorithm for turning the log. But it is getting stuck if the log is [tex]2\sqrt{2}(Width)[/tex]
 
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  • #2
the log will turn around the inner corner. just find a formula for finding the length of a line entirely in the water which passes through that point. then find the minimum
 
  • #3
Start by drawing a picture. Draw two channels of width 64 and 125 and draw a straight line from the outside edges just touching the inside corner. You should be able to find a formula for the length of that line in terms of x, the distance from the outside corner to one of the ends of the line. Then find x which minimizes length. You MIGHT be able to get that by completing the square but I seem to remember a problem like this requiring a derivative. Are you sure this is "precalculus"?
 

FAQ: Log Length in L-shaped River Channel

What is "Log Length" in an L-shaped river channel?

"Log Length" refers to the length of a log that is deposited or trapped in an L-shaped river channel. It is often used as a measure of the channel's hydraulic characteristics and can impact the flow and sediment transport within the channel.

How is "Log Length" measured in an L-shaped river channel?

Log length can be measured using a variety of techniques, including direct measurements of individual logs or using remote sensing methods such as aerial photography or LiDAR. The specific method used may vary depending on the size and complexity of the channel.

What factors influence "Log Length" in an L-shaped river channel?

Several factors can influence log length in an L-shaped river channel, including the amount and type of sediment in the channel, the velocity and direction of flow, and the presence of obstacles or other channel irregularities. Human activities such as land use changes and construction can also impact log length.

How does "Log Length" affect the ecology of an L-shaped river channel?

The length of logs in an L-shaped river channel can play a crucial role in creating and maintaining habitat diversity for aquatic organisms. Longer logs can provide shelter and food sources for various species, while shorter logs may contribute to flow and sediment patterns that support different types of vegetation.

Can changes in "Log Length" in an L-shaped river channel have ecological impacts?

Yes, changes in log length can have significant ecological impacts in an L-shaped river channel. Alterations to log length can affect the flow and sediment dynamics of the channel, alter the habitat and food sources for aquatic organisms, and impact the overall health and diversity of the ecosystem.

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