Question

A piece of string whose length is 32cm is cut into 2 pieces. one piece is used to form an equilateral triangle and the other to form of the circle so that the sum of the areas is a minimum? Find the minimum sum of the areas. Express your answer in pi. What is the function

Computation of derivative

Computation of Critical numbers

Computation of minimum value

Answer 1

Solution.

Determine:

x cm - a piece is used to form an equilateral triangle.

32-x cm - a piece is used to form a circle.

$S_1=\frac{a^2\sqrt 3}{4}$ - the area of the triangle.

$a=\frac x3$ ; $S_1=\frac{x^2\sqrt 3}{36}$.

$S_2=\pi r^2$ - the area of the circle.

$l=2\pi r$ ; $r=\frac {l}{2\pi}=\frac{32-x}{2\pi}$;

$S_2=\pi (\frac{32-x}{2\pi})^2=\frac {1}{4\pi}(32-x)^2$.

The sum of the areas:

$S=S_1+S_2= \frac{x^2\sqrt 3}{36}+ \frac {1}{4\pi}(32-x)^2$ .

Derivative of the function:

$S’=\frac{2x\sqrt 3}{36}-\frac{32-x}{2\pi}=0$

$\frac{2x\sqrt 3}{36}=\frac{32-x}{2\pi}$

$\frac{ x\sqrt 3}{9}=\frac{32-x}{\pi}$

$x_0=\frac{9\cdot32}{\pi\sqrt 3+9}$ - critical number.

As $S’’=\frac {\sqrt 3}{18}+\frac {1}{2\pi}>0$ so $S(x_0)$ is the minimum value of the function $S(x)$.

$S(x_0)=(\frac{9\cdot 32}{\pi\sqrt 3+9})^2\cdot\frac{\sqrt 3}{36}+\frac{1}{4\pi}(32-(\frac{9\cdot 32}{\pi\sqrt 3+9})^2)^2=$

$\frac{256}{\pi+3\sqrt 3}$

Answer: $S_{min}= \frac{256}{\pi+3\sqrt 3}$ $cm^2$ .