Question

The area of a rectangle is 3,246 m^2. Find the dimension of the rectangle that gives the smallest perimeter for the rectangle

Answer 1

\[Perimeter = 2*lengthOfsideA + 2*lengthOfsideB\] that is your constraint

Answer 2

you have \[A=3246\] which is \[A=lengthOfsideA*lengthOfsideB\]

Answer 3

a.b = 3246

2(a + b) should be min.

Answer 4

Use the equation for Area to find a value for one of the sides in terms of the other e.g. \[lengthOfsideA = \frac{A}{lengthOfSideB}\]

Answer 5

\[P=2(A+B)\] as Ishaan suggests and plug in the relation you discovered using the Area forumla, and then find the minimum of the function

Answer 6

What are you saying?

Answer 7

You do not know the lengths of either side of the rectangle

Answer 8

You do know the area, and you do know the formula to get area

Answer 9

you also know the formula for the perimeter of a rectangle

Answer 10

ughhh. I dont get it

Answer 11

a+b = 1623

Answer 12

3246 is the area: area is length of side A * length of side B in the rectangle

Answer 13

so \[3246 = A x B\]

Answer 14

\[3246=A*B\] and you have \[Perimeter = 2*A + 2*B\]

Answer 15

now, you want to use these two relationships

Answer 16

so determine the length of A with regard to the area by dividing the area by the length of B \[Area=3246=A*B\] now \[A=\frac{3246}{B}\]

Answer 17

OK, great, so you have \[A=\frac{3246}{B}\] and \[Perimeter = 2* A + 2*B\]

Answer 18

now you can substitute for A in the Perimeter equation because you found that A is \[\frac{3246}{B}\]

Answer 19

\[Perimeter = 2* (\frac{3246}{B}) + 2*B\]

Answer 20

now, we can convert this into a quadratic polynomial like \[a*x^2 + b*x + c=0\] with a little manipulation

Answer 21

first

Answer 22

move Perimeter to the right side
\[0 = 2*(\frac{3246}{B}) +2B - Perimeter\]then,
multiply the entire thing by \[B\] because the B in the denominator \[\frac{3246}{B}\] is nasty and harder to deal with than this
\[B*[2*(\frac{3246}{B})+ 2B - Perimeter] = \] is now \[2*(3246)+2B^2 - Perimeter*B=0\]

Answer 23

rearrange this to be \[2B^2 - Perimeter*B + 2*3246=0\]

Answer 24

omg. what was that? What is this called this is impossible

Answer 25

whats this process called?

Answer 26

this is just some substitution and algebra

Answer 27

Okay. well.. I feel like i took two steps forward and five steps back

Answer 28

check this out: if you have a rectangle with Area of 3246, if side A is 1 then side B would have to be 3246, that is super long and skinny rectangle

Answer 29

the perimeter on that is \[2*3246+2*1\]

Answer 30

that is huge

Answer 31

so imagine you are manipulating the sides of the rectangle, like when you can draw a shape in photoshop and move your cursor around to change the dimensions

Answer 32

you made a super long and skinny rectangle that has a huge perimeters, 3246 by 1, so you should probably squash it a bit, make it fatter

Answer 33

is this better for conceptualizing what is going on?

Answer 34

okay..

Answer 35

go on..

Answer 36

right, so you have the lengths of your rectangle

Answer 37

A and B

Answer 38

Perimeter of a rectangle is \[P=2*A + 2*B\] while \[Area=A*B\]

Answer 39

so, if you are squashing the super long rectangle such that the area still 3246 but the lengths of the sides are changing, ultimately, what shape do you think will yield the shortest sides?

Answer 40

I already got the answer. 56.97 by 56.97. Thanks