Q. 16 Given that 鈭哋AB聽is an isoscel... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 8: Q. 16 (page 556)

Given that $鈭� O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Expert verified

The area is $17.22$ square units.

Step by step solution

Step 1. Given Information

The given figure is

Step 2. Find the Angles of Triangle

The triangle is an isosceles triangle.
Sum of the angles of a triangle is $180 掳$ .
So,

$\begin{array}{rcl} 40 掳 + x + x & = & 180 掳 \\ 2 x & = & 140 掳 \\ x & = & 70 掳 \end{array}$

Step 3. Use rule of Sines

Use the rule of sines to find the missing side, which is also the diameter of the semicircle.

$\begin{array}{rcl} \frac{\sin 40}{a} & = & \frac{\sin 70}{5} \\ a & = & 5 \frac{\sin 40}{\sin 70} \\ = & 5 脳 0.68 \\ = & 3.42 \end{array}$

Step 4. Find the total area

The total area of the figure is the sum of the area of the triangle and the semicircle.
To find the area of the triangle, find the semiperimeter.

$s = \frac{3.42 + 5 + 5}{2} = 6.71$

So, the area of the triangle is calculated below.

$\begin{array}{rcl} A_{T} & = & \sqrt{6.71 脳 1.71 脳 1.71 脳 3.29} \\ = & 8.03 \end{array}$

And the area of the semicircle is calculated as shown below:

$A_{C} = 蟺 {(\frac{3.42}{2})}^{2} = 9.19$

So, the total area is $8.03 + 9.19 = 17.22$ square units.

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91影视

Given that $鈭� O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Angles of Triangle

Step 3. Use rule of Sines

Step 4. Find the total area

One App. One Place for Learning.

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91影视

Given that 鈭�OABis an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Angles of Triangle

Step 3. Use rule of Sines

Step 4. Find the total area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Geometry

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Given that $鈭� O A B$ is an isosceles triangle and the shaded sector is a semicircle, find the area of the entire region. Express your answer as a decimal rounded to two places.