Help! Advanced Algebra Matters... READ IT BELOW?

Question

(taken from the book)

A wire 36 m long is cut into 2 pieces. Each piece is bent to form a rectangle which is 1 cm longer than it is wide. How long should each piece be to minimize the sum of the areas if the two rectangles?

Answer 1

I am calling a & a + 1 the width and length of one rectangle and b & b + 1 the width and length of the other rectangle
A = total area of both

A = a(a+1) + b(b+1)

4a + 2 + 4b + 2 = 36
a = 8 - b

A = (8-b)(9-b) + b(b+1)
A = 2b² - 16b + 72
A = 2(b² - 8b) + 72
A = 2(b² - 8b + 16) + 72 - 32
A = 2(b - 4)² + 40

The minimum is at the vertex (4,40)

so b = 4
a = 8 - b = 4

So both rectangles are 4 x 5

Answer 2

i cannot be bothered to answer this completely but will set you off on the right track, providing your able to do the number crunching of calculus should be enough..

call one piece x meters long and the other (36 - x) meters long

then think of the rectangles formed - the one made from x will have length y and width (y-1)cm
2y + 2(y-1) = x

formulate an expression for the other rectangle using (36 - x) instead of x

sumthing like

2z + 2(z-1) = 36 - x

then find an expression for area of each, and then both added together

you will have to find out y in terms of x and same for z (once done this replace z with y appropriately)

then differentiate to find minimum of AREA = ax^2 + bx + C


GOOD LUCK

Answer 3

Okay letting x and y be the shorts sides of each rectangle,
then 2x+2(x+1)+2y+2(y+1)=36.
Or, x+x+1+y+y+1=18.
Or, 2x+2y=18-1-1=16.
SO, x+y=8 or x=8-y.
Now, we want to minimize x(x+1)+y(y+1).
Substituting, ((8-y)(8-y+1))+(y(y+1))=
((8-y)(9-y))+y^2+y.
Or, 72-8y-9y+y^2+y^2+y =
72-16y+2y^2.
Now differinate and set to 0,
-16+4y=0.
SO, y=4.

SO, both rectangles should be 4 by 5.

Answer 4

Recheck your problem statement. If you require all the wire to be used for 2 pieces and both rectangles are the same size, you don't have a minimum problem.

Answer 5

He sounds like a genius. no longer the modernized term that suggests he's overly wise the two. definitely a genius. Albert Einstein did terrible in college and look the place he was. My brother is a genius and yet does poorly in college. that's style of in basic terms a trait for geniuses that i be attentive to. do no longer obsess along with his corporation paintings, do no longer brag or boast, and don't hover over him 24/7. i'm no longer attempting to assist you to be attentive to the way you could nonetheless guardian yet definitely, with the intention to make him bored of the sector in classes and have confidence this is extensive-unfold and annoying inflicting him to slack off. enable him be a baby yet only attempt to slowly progression in sitting down and patiently getting to know with him. :)