Multivariable Calculus Maximum & Minimum
mahddh
I just finished a section in my calculus book that covered computing
Relative Maximum and Minimum extremas using partial derivatives.
There were two problems in the exercises that I do not know how to
work
Problem 1
Find 3 positive numbers that satisfy both of the following
conditions:
i. Their sum is 27
ii. The sum of the squares is as small as possible.
Well, I know I'll have this equation:
x+y+z=27
But, that's as far as I can get.
The answer is 9,9,9.
Problem 2
A shipper limits size of packages they will accept to a maximum
length + girth of 84 inches. Girth is the distance around middle of
the package (2x + 2z where x is width and z is height).
Determine the dimensions (x, y, z) of largest volume package that can
be shipped.
Again, I can get one equation and then I'm stuck.
2x + 2z + y = 84
The answer is supposed to be 14 inches wide, 28 inches long an 14
inches high.
Thanks in advance to all who respond.
Emanuel Meron
mahddh wrote:
> Problem 1
> Find 3 positive numbers that satisfy both of the following
> conditions:
> i. Their sum is 27
> ii. The sum of the squares is as small as possible.
> Problem 2
>
> A shipper limits size of packages they will accept to a maximum
> length + girth of 84 inches. Girth is the distance around middle of
> the package (2x + 2z where x is width and z is height).
>
> Determine the dimensions (x, y, z) of largest volume package that can
> be shipped.
problem 1:
Your numbers are x, y, z with x+y+z = 27
>>> z = [27-(x+y)]
Sum of squares function:
f(x,y) = x^2+y^2+[27-(x+y)]^2 = expand, etc... =
= 2x^2+2y^2-54x-54y+2xy+27^2
Partial derivatives:
df/dx = 4x+2y-54
df/dy = 2x+4y-54
Solve simultaneously:
4x+2y-54 = 0
2x+4y-54 = 0
x=9, y=9, 27-(x+y)=z=9
Problem 2: see above
Good luck
Gary Scott Simon
In article <343473...@netvision.net.il>, Emanuel Meron
<ms_...@netvision.net.il> wrote:
>mahddh wrote:
>> Problem 1
>> Find 3 positive numbers that satisfy both of the following
>> conditions:
>> i. Their sum is 27
>> ii. The sum of the squares is as small as possible.
>Your numbers are x, y, z with x+y+z = 27
>
> z = [27-(x+y)]
>
>Sum of squares function:
>
>f(x,y) = x^2+y^2+[27-(x+y)]^2 = expand, etc... =
> = 2x^2+2y^2-54x-54y+2xy+27^2
>
>Partial derivatives:
>
>df/dx = 4x+2y-54
>df/dy = 2x+4y-54
>
>Solve simultaneously:
>
>4x+2y-54 = 0
>2x+4y-54 = 0
>
>x=9, y=9, 27-(x+y)=z=9
If, during my first term at college, my calculus of several variables
instructor had ever explained anything this cogently, I'd probably be a
mathematician today.