maximum/minimum word problems

[nk]

could i get some help with this please?

a 30 cm piece of wire is cut in two. one piece is bent into the shape of a square, the other piece into the shape of a rectangular with a length to width ratio of 2:1. what are the lengths of the two pieces if the sum of the area of the square and the rectangle is a minimum?

 

[skeeter]

let x = length of the piece that forms the rectangle

then 30-x = length of the piece that forms the square

easy part first ... 30-x is the perimeter of the square, so each side of the square
is (30-x)/4. the area of the square is (30-x)²/16.

the rectangle has a length to width ratio of 2:1, or L = 2W ... since the perimeter of the rectangle is x

x = 2(L + W)
x = 2(2W + W) = 2(3W) = 6W

W = x/6, L = 2(x/6) = x/3

so, the area of the rectangle is L*W = (x/3)(x/6) = x²/18.

the sum of the two areas is ...

A = (30-x)²/16 + x²/18

A = (900 - 60x + x²)/16 + x²/18

A = 225/4 - 15x/4 + x²/16 + x²/18

A = 225/4 - 15x/4 + 17x²/144

the minimum area will be located at the vertex of the parabolic graph of this area equation ...

x = -b/(2a) = (15/4)/(17/72) = 270/17 = approx 15.9 cm

the other piece that forms the square would be 30-x = approx 14.1 cm in length

 

[nk]

im confused when youre multiplying and dividing in here, can you put the " * " character when its multiplying and " / " where its dividing. in the beginning where it becomes 30 - x squared times 16 there's " / " signs there. im confused

 

[stapel]

im confused when youre multiplying and dividing in here, can you put the " * " character when its multiplying and " / " where its dividing. in the beginning where it becomes 30 - x squared times 16 there's " / " signs there. im confused

And readers aren't confused by your lack of standard capitalization, punctuation, grammar, etc...? :?

There doesn't appear to be any multiplication in what is posted. And, by this stage in your mathematical career, you should have become comfortable with "multiplication by juxtoposition". That is, given "2x", you shouldn't need to have somebody explain that this mean "two times exx".

The division already is indicated by the division (or fraction) bar.

Thank you.

Eliz.

 

[skeeter]

im confused when youre multiplying and dividing in here, can you put the " * " character when its multiplying and " / " where its dividing. in the beginning where it becomes 30 - x squared times 16 there's " / " signs there. im confused

there is no 30 - x squared times 16 ... it's 30 - x squared divided by 16, just as I typed it.

 

[Denis]

ab means a times b (multiplication defaults) , a/b means a divided by b;
GET USED to it, it's STANDARD :shock: