Chapter 3.3, Problem 1MRPS

Solution

(a)

Write a functions $s\left(x\right)$ for the area of the square.

  $s\left(x\right)=x^2$

Given:

A circle is inscribed in a square as, as shown.

  

Concept Used:

• The composition function $f\circ g$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

According to given diagram

The side of square = $x$

The area of the square $\begin{array}{l}\\ s\left(x\right)=x\cdot x\\ =x^2\end{array}$

(b)

Write a functions $c\left(x\right)$ for the area of the circle.

  $c\left(x\right)=\frac{11x^2}{14}$

Given:

A circle is inscribed in a square as, as shown.

  

Concept Used:

• The composition function $g\circ f$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

In given diagram

The radius of circle $r=\frac{x}{2}$

Area of the circle = $c\left(x\right)$

  $\begin{array}{c}c\left(x\right)=\pi r^2\\ =\frac{22}{7}{(}^{\frac{x}{2}}\\ =\frac{22x^2}{7\cdot 4}\\ =\frac{11x^2}{7\cdot 2}\\ =\frac{11x^2}{14}\end{array}$

(c)

Write and simplify a functions $r\left(x\right)$ for the area of the shaded region.

  $r\left(x\right)=\frac{3x^2}{14}$

Given:

A circle is inscribed in a square as, as shown.

  

Concept Used:

• The composition function $g\circ f$ can be found by replacing x by $g\left(x\right)$ in the function $f\left(x\right)$
• The domain of a function $y=f\left(x\right)$ is the set of all x values for which the function is defined.
• A rational function $y=f\left(x\right)$ is undefined for the values of x , for which the denominator is zero.
• A radical function is defined for positive values greater than or equal to zero.
• Zero product property: if $a\cdot b=0\text{then}a=0\text{or}b=0$

Calculation:

In given diagram

The side of square = $x$

The radius of circle $r=\frac{x}{2}$

Area of the shaded region $r\left(x\right)$ = Area of the square $s\left(x\right)$ - Area of the circle $r\left(x\right)=s\left(x\right)-c\left(x\right)$

  $\begin{array}{c}r\left(x\right)=s\left(x\right)-c\left(x\right)\\ =x^2-\frac{11x^2}{14}\\ =\frac{14x^2-11x^2}{14}\\ =\frac{3x^2}{14}\end{array}$