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The logo for a new company is a circle divided into 21 same-sized sectors. Three of the sectors are white, and the others are different colors.

The area of one of the white sectors is 13π/54 inches squared.

What is the radius of the circle? Enter your answer in the box rounded to the nearest hundredth.

 Apr 16, 2024
 #1
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Here's how to find the radius of the circle:

 

Area of a Single White Sector:

 

We are given that the area of one white sector is 13π/54 square inches.

 

Proportion of White Sectors:

 

Since there are 21 sectors total and 3 are white, the white sectors represent 3/21 of the entire circle's area.

 

Full Circle Area:

 

To find the total area of the circle, we can divide the area of one white sector by its proportion (3/21) of the whole circle:

 

Total circle area = (Area of one white sector) / (Proportion of white sectors)

 

Total circle area = (13π/54 in²) / (3/21)

 

Circle Area Formula:

 

We know the area of a circle is calculated by πr², where r is the radius.

 

Equating Areas:

 

Since the total area we found represents the entire circle, we can equate it to the circle area formula:

 

πr² = (13π/54 in²) / (3/21)

 

Simplifying and Solving:

 

Cancel out common factors of π and simplify:

 

r² = (13 / 54) * (21 / 3) in²

 

r² = 13 in²

 

Take the square root of both sides (remembering that squaring can introduce extraneous solutions, so we'll check for those later).

 

r = √13 in ≈ 3.61 in (rounded to nearest hundredth)

 Apr 17, 2024

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