A rectangular field is 130 yards long and 85 yards wide.
Give the length and width of another rectangular field that has the same perimeter but a larger area.

Question

Florisalreadytaken · Answer

oopsies, lapsus again.... -- you are asked to find a rectangle with the same perimeter, but a LARGER area.

to get a rectangle with the same perimeter, but a larger area, we have to take something off the length, and and add the same reduced ammount to the width of it, so you will end up with a vertically-longer rectangle, and dimmed on the sides.

Florisalreadytaken · Answer

the width has got to be < than 130. but  > than 85.

lets take an example, which you can use as an answer. 
$ 115 $ long and $ 100   $ wide

$P_1=170+260 \Rightarrow P_1=430$
$P_2=200+230 \Rightarrow  P_2=430$

$A_1=130 \times 85 \Rightarrow  A_1=11050$
$A_2=115  \times 100 \Rightarrow A_2=11500$

you can see that there is a difference, and out rectangle is BIGGER.

surjithayer · Answer

@surjithayer wrote:

Perimeter =2(130+85)=2 \times 215=430 yards. Let the dimensions of new field be x and y. then 2(x+y)=430 x+y=430/2=215 y=215-x $$area~ A=xy=x(215-x)=215x-x^2$$ $$\frac{ dA}{dx }=215-2x$$ $$\frac{ dA}{ dx }=0,$$ gives 215-2x=0 2x=215 x=215/2=107.5 $$\frac{ d^2A }{ dx^2}=-2 $$ at x=107.5,$$\frac{ d^2A}{ dx^2 }=-2<0$$ Hence A is maximum at x=107.5 yards and y=210-107.5=107.5yards.

surjithayer · Answer

the rectangle has max. area if it is a square of each side=107.5 yards

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