Chapter 11.6, Problem 19PSC

a.

To determine

To calculate: The area of figure with sector and triangle.

Answer to Problem 19PSC

The area of figure is $50\pi -50\sqrt{3}$ .

Explanation of Solution

Given information:

Anequilateral triangle with side 10.

The center of each arc is the opposite vertex of equilateral triangle.

Formula used:

Area of equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

s = side of equilateral triangle.

Area of sector $=\left(\frac{measureofarc}{360^o}\right)\pi r^2$

r = radius of a circle.

Calculation:

  

  $\begin{array}{l}{A}_{sector}=\left(\frac{measureofarc}{360^o}\right)\pi r^2\\ $ =(\frac{{60}^{\circ }}{360^o})\pi {(10)}^2$=\frac{1}{6}\ast 100\pi \\ =\frac{50}{3}\pi \end{array}$

Area of equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

  $\begin{array}{l}A=\frac{{(10)}^2}{4}\sqrt{3}\\ A=25\sqrt{3}\end{array}$

Area of segment = Area of sector − Area of equilateral triangle

  $=\frac{50}{3}\pi -25\sqrt{3}$

There are 3 segments.

  $\begin{array}{l}{A}_{Total}=3\left(Areaofsegment\right)+A_{Triangle}\\ =3(\frac{50}{3}\pi -25\sqrt{3})+25\sqrt{3}\\ =50\pi -75\sqrt{3}+25\sqrt{3}\\ =50\pi -50\sqrt{3}\end{array}$

b.

To determine

To find: The perimeter of figure withthe help of circumference.

Answer to Problem 19PSC

The perimeter of figure is $10\pi$ .

Explanation of Solution

Given information:

An equilateral triangle with side 10.

The center of each arc is the opposite vertex of equilateral triangle.

Formula used:

Circumference of a circle: $C=2\pi r=\pi d$

r = radius of a circle.

d = diameter of a circle.

Calculation:

  

  $\begin{array}{l}C=\pi d\\ C=\frac{1}{6}*20\pi \\ C=\frac{10}{3}\pi \end{array}$

There are 3 segments.

  $\begin{array}{l}P=3*C\\ =3*\frac{10}{3}\pi \\ =10\pi \end{array}$