LaddiusMaximus
Find the dimensions of the rectangle of maximum area that can be formed from a 330-in. piece of wire
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cuddlepony
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YAY Optimization :)
cuddlepony
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so we have a rectangle
|dw:1334016584546:dw|
A = b * h
and we know that
b + h = 330 inches
Follow?
cuddlepony
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Now we need to solve for one of the variables
b = (330 - h)
thus
A = (330 - h)h
How about a little interaction :l
LaddiusMaximus
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so is that a standard formula?
cuddlepony
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This is normally what you do yes
LaddiusMaximus
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Sorry my son is crying.
cuddlepony
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its fine :)
LaddiusMaximus
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is that all there is to it?
cuddlepony
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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
cuddlepony
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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
cuddlepony
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Do you follow so far?
LaddiusMaximus
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165?
cuddlepony
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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
cuddlepony
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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
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cuddlepony
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sub a number smaller than 165 and a number larger than 165 into the function
A'(x) = 330 - 2h
cuddlepony
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Do this on your test for more marks :)
cuddlepony
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then check to see if it is positive or negative and put it in the square of the chart
LaddiusMaximus
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so 330-2(164)?
cuddlepony
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Keep it simple !!! I would go 330 - 2(0)
cuddlepony
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= 330 this is positive
f'(x) > 0
Function is increasing so
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cuddlepony
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Remember you can use any number smaller (unless it is not defined by the domain) :)
cuddlepony
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Fill in the next part of the chart :)
LaddiusMaximus
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330-2(1)?
LaddiusMaximus
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or 330-2(166)?
cuddlepony
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hell I will do it I should be studying General intermediary metabolism :) FML
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cuddlepony
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yup you can tell that the function will give you a negative number right
330 - 2(166)
cuddlepony
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thus you can deduce that your graph looks something like this|dw:1334017724724:dw|
cuddlepony
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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?
cuddlepony
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do you follow? If you dont understand something ask
LaddiusMaximus
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do i use base times heighy?
cuddlepony
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|dw:1334017945453:dw|
cuddlepony
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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
cuddlepony
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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
cuddlepony
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solve for b to find the length of the base
LaddiusMaximus
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0h 330-165. sorry Im way behind in this class. Its been a rough semester.
cuddlepony
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Do you see how obvious the answer is :) I hear you, I'm probably going to fail some of my classes :(
cuddlepony
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Remember when deal with these questions though always take into consideration each side of the object
cuddlepony
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so what would the actual lengths of h and b, be
LaddiusMaximus
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165 isnt right
cuddlepony
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I even explained how to get the final question earlier on lol. Think about it
cuddlepony
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you have
b= 165 inch
and
h = 165 inch
and you have a 4 sided rectangle
cuddlepony
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I went a little more of an indirect route because it kept the arithmatic simpler
cuddlepony
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The actual formula would be
2b + 2h = A
you cant make a 4 sided object with only 2 strands of wire
cuddlepony
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165/2 = 82.5 inches
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cuddlepony
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for all sides
cuddlepony
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To get the max area of a rectangle all sides have to be the same length :)
cuddlepony
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sorry for over complicating it but you would get the same answer if you used 2b + 2h = Area
LaddiusMaximus
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82.5 is not right either.
cuddlepony
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says who
cuddlepony
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82.5* 4 = 330
LaddiusMaximus
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webworks
cuddlepony
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Webworks is wrong
cuddlepony
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To maximize the area of a rectangle all sides must be of equal length
LaddiusMaximus
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dimension of maximum area? 82.5 in^(2)?
cuddlepony
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yes you probably forgot units
LaddiusMaximus
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it doesnt require me to enter units
cuddlepony
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well I recommend email the people overseeing your course, but lets ask someone to check first
LaddiusMaximus
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like who?
cuddlepony
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Someone else on the site
cuddlepony
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The answer I got is h = 82.5 and b = 82.5
xokatexo
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thats right
cuddlepony
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Yeah I know lol
cuddlepony
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You sadisfied Laddius?
cuddlepony
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satisfied*
LaddiusMaximus
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i suppose
cuddlepony
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do you understand the process?
cuddlepony
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Trust me it is right, logically it makes sense
cuddlepony
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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
cuddlepony
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The answer would be 85.5*85.5 or the Max Area sorry
cuddlepony
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the real answer would be 7310.25
cuddlepony
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sorry that is the answer I'm not use to using a calculator for these kinds of questions
cuddlepony
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try it I bet it works
LaddiusMaximus
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nope
LaddiusMaximus
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6806.25