It looks like the boundaries of
are the lines
and
, as well as the hyperbolas
and
. Naturally, the domain of integration is the set

By substituting
and
, so
, we have

and

so that

Compute the Jacobian for this transformation and its determinant.

Then the area element under this change of variables is

and the integral transforms to

Now compute it.

Answer:
Step-by-step explanation:
Let the rectangle have (x,y,z) as vertex in positive octant. The rectangular box has to be necessarily symmetrical about all the three axes.
Then the sides of the box would be

Volume = 
Maximize volume subject to

i.e. 
Use Lagrangian multipliers , we have
at the maximum


Dividing we get

Similarly 
Thus we get 
Hence dimensions are
(2x,2y,2z)
So dimensions are

Answer: 
Step-by-step explanation:
Given: Density of tank = 0.4 fish over feet cubed.
We know that, 
Therefore,

Hence, the volume of the tank = 
x= 15
4x+8x: 180
12x: 180
x: 180/12
x: 15
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