Density Functions - Oceanography

[xr250h]

Homework Statement

http://img39.imageshack.us/img39/3143/calcquiz3.jpg

The writing says: A(x)=int(p(y),y=0..x) <-- Maple format

Homework Equations

A(x)=int(p(y),y=0..x)

The Attempt at a Solution

So I did not attend class the day we went over this and he only went over it that day. I honestly have no idea how to even go about doing it, but here's a stab in the dark:

I believe the "water column" he is talking about is a rectangular prism with length=1 m, width=1 m, and height=50 m.

The density function is p(x)=10 cod per cubic meter (where p is "rho" and x is an area?)

a. Total number of cod in water column- Wouldn't this be the area of the column * 10?

So, A=LWH=50 m^3. 50*10=500 cod. Also, this is the same answer I would get if I took the Area under the curve p(x)=10 from x=0..10.

For the other functions, would I take the area under the curve as well?

b. Depth where fish density is largest - Would this be the maximum point on the graph? If so, the first one would be 0<x<50 because the density of fish is constant.

c. Number of cod between 20 and 30 meters - Simply the area under the curve from 20 to 30?


Any help is much appreciated!

Thanks,
Ryan

 

[mighty2000]

Dear Ryan,

It seems like you have a good understanding of the problem and the concepts involved. Let's break down the problem and go through it step by step.

First, let's define the variables and units involved:
- A(x) represents the total number of cod in the water column up to a certain depth x (in meters).
- p(y) represents the density of cod at a certain depth y (in cod per cubic meter).
- x represents the depth in meters.
- y represents the height in meters.
- The dimensions of the water column are 1m x 1m x 50m.

Now, let's look at the equation A(x)=int(p(y),y=0..x). This is a definite integral, which means it represents the area under the curve of the function p(y) from y=0 to y=x. In other words, it is the total number of cod in the water column up to a certain depth x, since p(y) represents the density of cod at a certain depth y.

a. To find the total number of cod in the water column, we use the equation A(x)=int(p(y),y=0..x). Since p(y) is a constant function (10 cod per cubic meter), the integral becomes A(x)=10x. Substituting x=50 (since the water column is 50m deep), we get A(50)=10(50)=500 cod.

b. The depth where the fish density is largest can be found by looking at the graph of p(y). Since p(y) is a constant function, the density is the same at all depths. Therefore, the maximum point on the graph is at any depth between 0 and 50 meters.

c. To find the number of cod between 20 and 30 meters, we use the equation A(x)=int(p(y),y=0..x). We want to find the area under the curve of p(y) from y=20 to y=30. This can be done by subtracting the area under the curve from y=0 to y=20 from the area under the curve from y=0 to y=30. In other words, A(30)-A(20)=int(p(y),y=0..30)-int(p(y),y=0..20). Since p(y) is a constant function, this becomes 10(30)-10(20)=