# Find the dimensions of a rectangle with area 512 m^2 whose perimeter is as small as possible. ?

(If both values are the same number, enter it into both blanks.)

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• 2 weeks ago

A rectangle with an area of 512 m^2

whose perimeter is as small as possible is a

a square which measures 16√2 m on a side (approx 22.627 m)

• RR
Lv 7
2 weeks ago

It has to be a square.

Try it with 16 square m

it could be 4 x 4 (perimeter 4 + 4 + 4 + 4 = 16)

It could be 2 x 8 (perimeter 2 + 2 + 8 + 8 = 20)

it could be 1 x 16 (perimeter 1 + 1 + 16 + 16 = 34)

The 4 x 4 square has the smallest perimeter.

So all you need to do is find the square root of 512

sq rt 512 = 22.627M

It's a square 22.627 x 22.627

• ?
Lv 7
2 weeks ago

Area A = ℓ x w = 512 m² ⇒ ℓ = (512 m²)/w

Perimeter P(w) = 2ℓ + 2w = 2(512 m²)/w + 2w

................P(w) = 1024/w + 2w

...........................1024 + 2w²

................P(w) = --------------, w > 0

..................................w

// We have the restriction w > 0 for P(w)

// because you can't have w=0 in the denominator

// and you can't have a negative length

// Find the 1st derivative P '(w)

.............w [4w] - (1024 + 2w²)[1]

P '(w) = --------------------------------

..........................w²

.............4w² - 1024 - 2w²

P '(w) = ----------------------

..........................w²

.............2w² - 1024

P '(w) = ---------------

...................w²

// Find the critical values, where P '(w) = 0

P '(w) = 0 ⇔ 2w² - 1024 = 0

................⇒ 2w² = 1024

................⇒ w² = 512

................⇒ w = ± 8√2

// Find the 2nd derivative P "(x)

..............w² [4w] - (2w² - 1024)[2w]

P "(w)  = ----------------------------------

.........................(w²)²

..............2048w

P "(w) = ----------

................w⁴

// Apply the 2nd derivative test to the critical values.

When w = -8√2,

P "-(8√2) < 0 ⇒ P(w) has a relative MAX at w= -8√2

When w = +8√2

P "(8√2) > 0  ⇒ P(w) has a relative MIN at w = +8√2

// Therefore, the dimensions that result in a minimum perimeter are

w = +8√2 and

ℓ = 512/8√2 = 32√2................ANS

• 2 weeks ago

A square is the rectangle whose perimeter is the smallest for a given area.

l = w = √512

= √256√2

= 16√2

≈ 22.627417 m