When changes in one variable are usually accompanied by changes in the same direction in another variable?
1 answer:
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The curve 
is parameterized by
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
, we have gradient 
. The theorem itself then says that the line integral of 
along a curve 
parameterized by 
, where 
, is given by
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating
, a scalar function.
so in the line integral, we have
You are mistaken in thinking that the gradient theorem applies here. Recall that for a scalar function
Specifically, in order for this theorem to even be considered in the first place, we would need to be integrating with respect to a vector field.
But this isn't the case: we're integrating
Answer:
The package of maximum dimensions has:
Width and height = 21 in each , and length = 42 in
Step-by-step explanation:
A package of square cross section has a girth equal to the perimeter of the square ("4 x" if we consider "x" as the side of the square). This quantity, added to the package's length "L" has to be no more than 126 in.
The maximum allowed for priority mail is therefore:
At the same time, we want the volume of the package to be a maximum. The volume of this parcel is defined as the product of all three dimensions of the package:
We can now use the first formula for the priority mail maximum dimensions, to write L in terms of "x" and replace it in the volume formula:
So now the volume expression becomes:
If the student doesn't know calculus, a graphing tool can be used to find the maximum.
With calculus derivatives the maximum can be easily found:
We can request the derivative of the volume to satisfy the conditions for derivative = 0 and the function concave down to locate the function's "maximum".
This tells us that there are two solutions to the derivative equal zero:
The first solution is clearly a minimum since it would render a package of zero volume. the second solution () is the one that corresponds to the maximum of the volume function.
Therefore, the dimensions of the package of largest volume are: width and height: 21 in each, and length L= 126 - 4*21 = 42 in