105. A circular sector has an 8.26-inch radius and a 12.84-inch arc length. There is another sector that has the same area and the same perimeter. What are its measurements

As the radius of a sector decreases, the arc length increases to have an

equal area with a sector with a radius that is longer.

The measurements of another sector with the same perimeter and area are;

Reasons:

Radius of the sector, r = 8.26 inches

Arc length of the sector = 12.84 inches

Perimeter of sector = 2·r + Arc length

Perimeter of the given sector = 2 × 8.26 inch + 12.84 inch = 29.36 inch

Circumference of the circle = 2 × 8.26 × π = 16.52·π

Area of the given circle, A = π × 8.26² = 68.2276·π

Area of the sector, A = 53.0292 inch²

Therefore, we have;

2·R + A = 29.36 inches

A·R = 53.0292 × 2

Which gives;

R² + 53.0292 = 14.68·R

R² - 14.68·R + 53.0292 = 0

Factorizing with a graphing calculator, gives;

(R - 6.42)·(R - 8.26) = 0

R = 6.42 or R = 8.26

The measurement of R = 8.26 is for the first sector

Given that the sectors are different, we have;

The arc length, A = 29.36 - 2·R

∴ A = 29.36 - 2 × 6.42 = 16.52

Learn more about the sector of a circle here:

link

Area of another sector=

Perimeter of another sector=29.36 in

Step-by-step explanation:

We are given that

Radius, r=8.26 in

Arc length , l=12.84 in

Area of circular sector=

Area of circular sector=

Where Circumference of circle=

Substitute the values

Area of circular sector=

Area of circular sector=

Area of another sector=

Perimeter of circular sector=8.26+8.26+12.84

Perimeter of circular sector=29.36 in

Perimeter of another sector=29.36 in

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step-by-step explanation: