How Does Observer Motion Affect Detected Fish Concentration Changes?

In summary, the problem asks to find the total change in fish concentration detected by an observer moving away from a feeding point in a lake with a speed of 10 m/s, and the corresponding change detected by a stationary observer. The concentration of fish is represented by c(x,y) = 1/(x2 + y2), and by converting to polar coordinates, c(r) = 1/r2. The rate of change of concentration with respect to r can be calculated using the chain rule. With this method, the problem can be solved successfully.
  • #1
banerjeerupak
123
1

Homework Statement



The concentration c of fish away from a feeding point in a lake is given by c(x,y) = 1/(x2 + y2). Find the total change of fish concentration detected by an observer riding a boat traveling with a speed u = 10 m/s straight away from the feeding point. What is the corresponding change detected by a stationary observer.

Homework Equations





The Attempt at a Solution



Can someone give me how to start solving this problem. I'm having trouble making a start at it.
 
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  • #2
I'm guessing that the concentration of fish is actually
[tex]c(x, y)~=~\frac{1}{x^2 + y^2}[/tex]

Since the boat with the observers is moving in a straight line away from the feeding point, c(x, y) = c(r) = 1/r2. Can you calculate the rate of change of concentration with respect to r?
 
  • #3
Mark44's idea, to change to "polar coordinates" is excellent. To change from "an observer riding a boat traveling with a speed u = 10" to "a stationary observer", use the chain rule: dc/dt= dc/du du/dt
 
  • #4
Thank you very much... question solved.
 

Related to How Does Observer Motion Affect Detected Fish Concentration Changes?

1. What is the formula for calculating rate of change?

The formula for calculating rate of change is (change in y)/(change in x), or (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on a line.

2. How do you interpret the rate of change?

The rate of change represents the slope of a line, which is the measure of how steep the line is. A positive rate of change indicates an upward slope, while a negative rate of change indicates a downward slope.

3. What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is calculated over a specific interval, while the instantaneous rate of change is calculated at a specific point. In other words, the average rate of change represents the overall change over a given time period, while the instantaneous rate of change represents the change at a single moment.

4. How can rate of change be used in real-world situations?

Rate of change can be used to analyze and understand various real-world phenomena, such as population growth, stock market trends, or the speed of a moving object. It can also be used to make predictions and forecasts based on past data.

5. What are some common units for rate of change?

The units for rate of change depend on the variables involved. For example, if the variables are time and distance, the units for rate of change would be distance per unit of time (such as meters per second). Some other common units include dollars per hour, percent per year, or pieces per minute.

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