Question

A piece of wire of length 80cm is to be cut into two sections. One section is to be bent into a square, and the other into a rectangle 4 times as long as it is wide.

(a) Let x be the side length of the square and y be the width of the rectangle. Write a formula connecting x and y and show that if A is the sum of the areas of the square and rectangle, then A = 41y^2 /4 − 100y + 400.

(b) Find the lengths of both sections of wire if A is to be a minimum.

Answer 1

I love optimization calculus problems! Alright, so first off show me what equations you think might be important here. Just say what you're thinking right now about it so I can help point you in the right direction.

Answer 2

hmm. Its quadratic function problem and i have no idea what the equation is

Answer 3

Well you know how long the wire is and it's going to be bent into shapes. That basically means you're going to have to have some kind of perimeter equation right? It also mentions area, so that's important.

Draw out a square and a rectangle and label the sides with variables, and pay attention to what they want, they seem kind of specific on what you call x and y.

Then write out equations for the perimeter of the square, of the rectangle, and the area of the square and the rectangle.

When you've gotten this far, try to take it further and see if you can connect these equations based on what you know or what might make it easier and ask yourself what the HELL am I actually trying to do. Take your time, once you understand optimization of one problem other problems start to become much easier and almost obvious.

Answer 4

Don't be afraid to kill trees in learning calculus. Draw pictures and equations and label things. The less stuff you keep juggling around in your mind, the more you can concentrate on what you're doing with them because the pieces are lying out in front of you.

Answer 5

so one wire is 80-x and other will be x as we dont know what length it has been cut ?

Answer 6

Yep.

Answer 7

actually ill use a instead of x as x is going to be used

Answer 8

Yeah, I was just about to say that haha, I haven't actually solved it yet, but I'm about to now.

Answer 9

this right so far?

Answer 10

Exactly right so far, keep it up.

Answer 11

Now how do we write a formula connecting x and y?

Answer 12

I don't want to give you the answer, but think about the perimeter. That's where you'll find your answer.

Answer 13

perimeter of square= a
perimeter of rectangle = 80-a
But it doesn't tell us which one is square or rectangle?

Answer 14

Perimeter of square= 4x
Perimeter of rectangle = 10y

Answer 15

What's 80 centimeters?

Answer 16

the sum of 2 perimeters

Answer 17

4x + 10y = 80

Answer 18

Perfecto

Answer 19

Continue with part (a), I think it shouldn't be too much more trouble.

Answer 20

ok i proved part (a) :)

Answer 21

Awesome, so now part (b). Do you understand what's going on when you're optimizing? If you have a graphing calculator I suggest you graph the area function and look at it and try to understand it. I'll help you if you need it, it's slightly confusing to understand at first.

Answer 22

Is there 2 answers as it said both sections? and if there is 2 answers how?

Answer 23

To save some possible confusion with your own calculator, I just graphed it online:

http://www.wolframalpha.com/input/?i=A%20%3D%2041y%5E2%20%2F4%20%E2%88%92%20100y%20%2B%20400.&t=crmtb01

Now you can see that it's a graph where the height represents the total area of both the rectangle and square. Since they are shapes related by not only their areas but their perimeters, you were able to make it a function of only one side of the rectangle which is pretty nifty.

So when you go from left to right, even though the graph doesn't show it, you can only go from 0 to 8. 0 means you didn't cut it and made a square (which is totally allowed in these types of problems some times!) and 8 means you also didn't cut it and you just bent it into a rectangle. Make sure this makes sense to you as a restriction on your domain for x.

Now, you want to know what gives you the smallest possible area for both shapes, and where is the smallest area? Well this is the function of area, and area is lowest right at the bottom of the parabola. We know that the slope=0 at that lowest point.

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I am about to go to sleep so I typed all that out generic stuff, let me answer your question.

Answer 24

There will be only one answer, since only one cut decides the length of both pieces of wire.

Answer 25

Well, you have to make sure you say which piece is the rectangle and which piece is the square, so yeah, you kind of have two answers.

Answer 26

ok thanks for your help ^^

nice graphing calculator LOL

Answer 27

Your answer will be in terms of the shortest length of the rectangle, y. So from there you can figure out the perimeter of the rectangle and then you know that's where you cut, the rest is the square.

If you need help understanding more I'll be here for about 20 minutes more, so you've got time.

Answer 28

i have gotten 200/41 as i used -b/2a to find the vertex

Answer 29

Wait is this a calculus class or an algebra class?

Answer 30

lol

Answer 31

It's right, I would just rather have seen you take the derivative lol.

Answer 32

Im in Australia and its a everything class... lol (year 11). I can take the derivative if you want lol

Answer 33

I mean, it's all good, it's just that calculus is cooler imo lol.

Answer 34

Taking the derivative and setting it equal to zero is exactly the same idea to find vertexes, but is a little more general since you can use that on cubic or whatever. Goodnight haha.