Chapter 2.6, Problem 11AYU

11. Geometry A wire 10 meters long is to be cut into two pieces. One piece will be shaped as a square, and the other piece will be shaped as a circle. See the figure.

(a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the square.

(b) What is the domain of A?

(c) Graph $A\text{ = }A\left(x\right)$. For what value of x is A smallest?

To determine

To find:

a. To express the total area $A$ enclosed by the pieces of wire as a function of the length $x$ of a side of the square.

Answer to Problem 11AYU

Solution:

a. $A=x^2+\frac{25+4x^2-20x}{\pi }$

Explanation of Solution

Given:

A 10m long wire is cut into two pieces. One piece is shaped as a square and other piece is shaped as a circle.

Calculation:

a. To express the total area $A$ enclosed by the pieces of wire as a function of the length $x$ of a side of the square.

From the given figure, the perimeter of the square$=4x$

Therefore, Area of the square $A_1=x^2$.

The perimeter of the circle $=2\pi r=\left(10-4x\right)$.

$r=\left(10-4x\right)/2\pi$

Area of the circle $A_2= \pi r^2={\left(10-4x\right)}^2/4\pi$.

Therefore, the total area enclosed by the wire $=A_1+A_2$.

$=x^2+\frac{{\left(10-4x\right)}^2}{4\pi }$

$=x^2+\frac{25+4x^2-20x}{\pi }$

To determine

To find:

b. To the domain of $A$.

Answer to Problem 11AYU

Solution:

b. $\text{Domain of}A=\left\{x|0<x<2.5\right\}$.

Explanation of Solution

Given:

A 10m long wire is cut into two pieces. One piece is shaped as a square and other piece is shaped as a circle.

Calculation:

b. To the domain of $A$:

Since, perimeter $\left(10-4x\right)$ has to be positive, the domain of

$A\left(x\right)=\left\{x|0<x<2.5\right\}$

To determine

To find:

c. To graph $A=A\left(x\right)$ and to find the value of $x$ is $A$ smallest.

Answer to Problem 11AYU

Solution:

c. $x=1.4$

Explanation of Solution

Given:

A 10m long wire is cut into two pieces. One piece is shaped as a square and other piece is shaped as a circle.

Calculation:

c. To graph $A=A\left(x\right)$ and to find the value of $x$ is $A$ smallest.

From the graph it can be seen that when $x=1.4,A$attains its minimum value $3.5$.

Therefore, $A$ is smallest when $x=1.4$.