Question

The rectangular bird sanctuary with one side along a straight river is to be constructed so that it contains 8 km^2 of area. Find the dimensions of the rectangle to minimize the amount of fence necessary to enclose the remaining three sides.

Answer 1

you want to apply calculus for this. have you learned calculus before?

Answer 2

Yes Can you help me derive an equation

Answer 3

yes. You can take the advantage of calculus at this point.
fist let's draw our sanctuaryd say it has x and y sides

Answer 4

Area(A) = xy = 8

Answer 5

sum of distance of fence(D) = 2x + y

Answer 6

now lets take D in terms of x using above 2 equations

Answer 7

D = 2x + 8/x

Answer 8

now this is the distance of fence. you need to find minimum of it. so we take first derivative of D with respect to x

Answer 9

ok

Answer 10

So
d(D)/d(x) = 2 - 8/x^2

Answer 11

and you will get
\[\frac{ d(D) }{ d(x) } = \frac{ 2(x-2)(x+2) }{ x^2 }\]

Answer 12

now draw the graph.

Answer 13

I lost you for how you got your equation for d/x

Answer 14

how is it 2-(8/x^2)

Answer 15

if put values for x from negative to positive. you will notice that
if x<-2 ; d(D)/d(x) is positive
if x = -2 ;d(D)/d(x) is zero
if -2 <x < 2 ;d(D)/d(x) is negative
if x = 2 ;d(D)/d(x) is zero
if x < 2 ;d(D)/d(x) positive

Answer 16

that's how I argue that fence has its minimum value when x = 2
and of course a distance cannot be an negative value anyway:). so we forget x = -2

Answer 17

now go back to our distance equation

D = 2x + 8/x

Answer 18

put x = 2
D = 2*2 + 8/2 = 4 +4 = 8

Answer 19

hope this helps

Answer 20

thanks!