Question

Find the dimensions of the rectangle with area 196 square inches that has minimum perimeter, and then find the minimum perimeter.

Answer 1

So find all the factors. Oh dis gon be hard... So just find all the factors right. Then add all the sums you know like look: 4 x 4 example if it equalled to 196, 4 +4 = 8. and then multiply by 2. so do every possibility. it takes time. then find the smallest sum

Answer 2

The minimum perimeter and dimensions of the rectangle is required.

The dimensions of the rectangle is 14 inches length and 14 inches breadth.

The minimum perimeter of the rectangle is 56 inches.

Let $x$ be the length of the rectangle.

$y$ be the breadth of the rectangle.

The area of the rectangle is $196\ \text{inch}^2$

Area is given by

$xy=196\\\Rightarrow y=\dfrac{196}{x}$

Perimeter is given by

$P=2x+2y\\\Rightarrow P=2x+2\times \dfrac{196}{x}\\\Rightarrow P=2x+\dfrac{392}{x}$

Differentiating with respect to $x$

$P'=2-\dfrac{392}{x^2}$

Equating to zero

$0=2-\dfrac{392}{x^2}\\\Rightarrow x=\sqrt{\dfrac{392}{2}}\\\Rightarrow x=14$

Double derivative of the function

$P''=\dfrac{2\times 392}{x^3}\\\Rightarrow P''=\dfrac{784}{x^3}$

Substituting $x=14$

$P''(14)=\dfrac{784}{14^3}=0.285>0$

Since, the value is greater than 0 perimeter will be minimum at $x=14$

Finding $y$

$y=\dfrac{196}{x}=\dfrac{196}{14}\\\Rightarrow y=14$

The minimum perimeter is

$P=2x+2y=2\times 14+2\times 14\\\Rightarrow P=56\ \text{inches}$

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