Find the dimensions of the rectangle of maximum area that can be formed from a 330-in. piece of wire

- LaddiusMaximus

Find the dimensions of the rectangle of maximum area that can be formed from a 330-in. piece of wire

- jamiebookeater

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- anonymous

YAY Optimization :)

- anonymous

so we have a rectangle
|dw:1334016584546:dw|
A = b * h
and we know that
b + h = 330 inches
Follow?

- anonymous

Now we need to solve for one of the variables
b = (330 - h)
thus
A = (330 - h)h
How about a little interaction :l

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## More answers

- LaddiusMaximus

so is that a standard formula?

- anonymous

This is normally what you do yes

- LaddiusMaximus

Sorry my son is crying.

- anonymous

its fine :)

- LaddiusMaximus

is that all there is to it?

- anonymous

So next step we need to take the derivative with A respect to the change in x
so
(Ad/dx) A = (Ad/dx) 330h - h^(2)
A' = 330 - 2h

- anonymous

Now We need to look at the domain of A' = 330 - 2h and see if we can find any places where it violates the domain (as this could be a place where the graph changes)
The domain is clearly all real numbers
Next We set
A'(x) = 0 (write this on your test it looks good and will get you a mark probably)
330 - 2h = 0
Now solve

- anonymous

Do you follow so far?

- LaddiusMaximus

165?

- anonymous

btw there is no standard formula but we know:
Area of a Square (or Rectangle) = Base * Height
We also know that the length of the wire is all sides added up
so we get
b + h = Length of wire
or we could express it as
2b + 2h = Length of wire
But really that would give you the same answer its best to keep it simple

- anonymous

h = 165 so we have a point
Now we need to use the rules
MEMORIZE THESE BTW
f'(x) > 0 The function is increasing
f'(x) < 0 The function is decreasing
Make a Table
|dw:1334017324830:dw|

- anonymous

sub a number smaller than 165 and a number larger than 165 into the function
A'(x) = 330 - 2h

- anonymous

Do this on your test for more marks :)

- anonymous

then check to see if it is positive or negative and put it in the square of the chart

- LaddiusMaximus

so 330-2(164)?

- anonymous

Keep it simple !!! I would go 330 - 2(0)

- anonymous

= 330 this is positive
f'(x) > 0
Function is increasing so
|dw:1334017519222:dw|

- anonymous

Remember you can use any number smaller (unless it is not defined by the domain) :)

- anonymous

Fill in the next part of the chart :)

- LaddiusMaximus

330-2(1)?

- LaddiusMaximus

or 330-2(166)?

- anonymous

hell I will do it I should be studying General intermediary metabolism :) FML
|dw:1334017660396:dw|

- anonymous

yup you can tell that the function will give you a negative number right
330 - 2(166)

- anonymous

thus you can deduce that your graph looks something like this|dw:1334017724724:dw|

- anonymous

Can you now see the answer?
h = 165
So now we know the max size of the height to give optimial area
Can you guess how to solve for the base?

- anonymous

do you follow? If you dont understand something ask

- LaddiusMaximus

do i use base times heighy?

- anonymous

|dw:1334017945453:dw|

- anonymous

No think about what you are doing, go back to the original formulas
Calculus is just measuring change.
With these problems we just are measuring how the Area changes when we change the length of h

- anonymous

Area of a rectangle = base * height
Length of The wire = Height length + Base length
A = b * h
and we know that
b + h= 330 inches
If we know that base length is = 165 and total wire used is = 330
165 + b = 330

- anonymous

solve for b to find the length of the base

- LaddiusMaximus

0h 330-165. sorry Im way behind in this class. Its been a rough semester.

- anonymous

Do you see how obvious the answer is :) I hear you, I'm probably going to fail some of my classes :(

- anonymous

Remember when deal with these questions though always take into consideration each side of the object

- anonymous

so what would the actual lengths of h and b, be

- LaddiusMaximus

165 isnt right

- anonymous

I even explained how to get the final question earlier on lol. Think about it

- anonymous

you have
b= 165 inch
and
h = 165 inch
and you have a 4 sided rectangle

- anonymous

I went a little more of an indirect route because it kept the arithmatic simpler

- anonymous

The actual formula would be
2b + 2h = A
you cant make a 4 sided object with only 2 strands of wire

- anonymous

165/2 = 82.5 inches
|dw:1334018556160:dw|

- anonymous

for all sides

- anonymous

To get the max area of a rectangle all sides have to be the same length :)

- anonymous

sorry for over complicating it but you would get the same answer if you used 2b + 2h = Area

- LaddiusMaximus

82.5 is not right either.

- anonymous

says who

- anonymous

82.5* 4 = 330

- LaddiusMaximus

webworks

- anonymous

Webworks is wrong

- anonymous

To maximize the area of a rectangle all sides must be of equal length

- LaddiusMaximus

dimension of maximum area? 82.5 in^(2)?

- anonymous

yes you probably forgot units

- LaddiusMaximus

it doesnt require me to enter units

- anonymous

well I recommend email the people overseeing your course, but lets ask someone to check first

- LaddiusMaximus

like who?

- anonymous

Someone else on the site

- anonymous

The answer I got is h = 82.5 and b = 82.5

- anonymous

thats right

- anonymous

Yeah I know lol

- anonymous

You sadisfied Laddius?

- anonymous

satisfied*

- LaddiusMaximus

i suppose

- anonymous

do you understand the process?

- anonymous

Trust me it is right, logically it makes sense

- anonymous

Its easy
1st acquire two formulas from the question
2nd Solve for one of the variables in one of the equations and sub it into the other equation
3rd take the derivative of the new equation you made
4th check domain and points the function is 0
5th look to see if function is increasing or decreasing and at what point it is at its max or min and that will be your answer
6th sub the answer back into the one of the equation to solve for the other variable.
7th check to see if your answer is reasonable

- anonymous

The answer would be 85.5*85.5 or the Max Area sorry

- anonymous

the real answer would be 7310.25

- anonymous

sorry that is the answer I'm not use to using a calculator for these kinds of questions

- anonymous

try it I bet it works

- LaddiusMaximus

nope

- LaddiusMaximus

6806.25

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