Q. B Suppose that you want to cut a r... [FREE SOLUTION]

Chapter 3: Q. B (page 314)

Suppose that you want to cut a rectangular shape with a particular area A from a sheet of material, and that you want the perimeter of the shape to be as small as possible. Use techniques of optimization to argue that the smallest possible perimeter will be achieved if the rectangular shape that you cut out is a square.

Short Answer

Expert verified

When $x = y$ , i.e. the length is equal to the width of the rectangular shape. Then the smallest possible perimeter will be achieved if the rectangular shape that you cut out is a square.

Step by step solution

Step 1. Given information.

Consider the given question,

Area of the rectangular shape is A.

Step 2. Find the critical numbers of function of perimeter.

Assume x,y to be the length and width of rectangular shape with area A.

$y = \frac{A}{x}$

Now, perimeter of the rectangle is given below,

$P = 2 (x + y) P = 2 x + \frac{2 A}{x}$

Differentiating with respect to x,

$\frac{d P}{d x} = 2 - \frac{2 A}{x^{2}}$

To find the critical numbers of function of perimeter,

$\frac{d P}{d x} = 0 2 - \frac{2 A}{x^{2}} = 0 x = \sqrt{A}$

Step 3. Determine the smallest possible perimeter.

Consider the obtained value of x,

$\frac{d^{2} P}{d x^{2}} = \frac{4 A}{x^{3}} \frac{d^{2} P}{d x^{2}} = \frac{4 A}{A^{\frac{3}{2}}} \frac{d^{2} P}{d x^{2}} = \frac{4}{\sqrt{A}} \frac{4}{\sqrt{A}} > 0$

That is $x = y$ .

Thus, the smallest possible perimeter will be achieved if the rectangular shape that is cut is square.

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the critical numbers of function of perimeter.

Step 3. Determine the smallest possible perimeter.

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