13. minimum - perimeter rectangles of all rectangles of area 100, which one has the minimum perimeter?

Explanation:

Step1: Define variables

Let the length of the rectangle be (x) and the width be (y). Given that the area (A = xy=100), so (y=\frac{100}{x}). The perimeter (P = 2(x + y)=2\left(x+\frac{100}{x}\right)), (x>0).

Step2: Find the derivative of the perimeter function

Differentiate (P(x)) with respect to (x). Using the sum rule ((u + v)^\prime=u^\prime + v^\prime) and the power rule ((x^n)^\prime=nx^{n - 1}), where (u = x) ((u^\prime=1)) and (v=\frac{100}{x}=100x^{-1}) ((v^\prime=- 100x^{-2}=-\frac{100}{x^{2}})). Then (P^\prime(x)=2\left(1-\frac{100}{x^{2}}\right)).

Step3: Find the critical points

Set (P^\prime(x) = 0), so (2\left(1-\frac{100}{x^{2}}\right)=0). Divide both sides by (2): (1-\frac{100}{x^{2}}=0). Then (\frac{100}{x^{2}}=1), which gives (x^{2}=100). Since (x>0), (x = 10).

Step4: Use the second - derivative test

Differentiate (P^\prime(x)) to get (P^{\prime\prime}(x)). (P^\prime(x)=2-\frac{200}{x^{2}}), and (P^{\prime\prime}(x)=\frac{400}{x^{3}}). When (x = 10), (P^{\prime\prime}(10)=\frac{400}{10^{3}}=\frac{400}{1000}=0.4>0). So the function (P(x)) has a minimum at (x = 10). When (x = 10), (y=\frac{100}{x}=10).

Answer:

The rectangle with length (x = 10) and width (y = 10) (a square) has the minimum perimeter.