A circle with a radius of 15 yards contains a sector with an area of 609 yde. Find the measure of the central angle of the sector in both radians and degrees.

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The measure of the central angle in degrees and radians are 310.32° and 5.42 rad respectively

where

r = radius

∅ = central angles

Therefore,

609 = ∅ / 360 × 3.14 × 15²

609 = 706.5∅ / 360

219240 = 706.5∅

∅ = 219240 / 706.5

∅ = 310.318471338

∅ = 310.32°

1° = 0.0174533 rad

310.32° = ?

Angle in radian = 310.32° × 0.0174533  = 5.41610573 ≈ 5.42 rad  

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310.3° or 5.41 radians.

Step-by-step explanation:

The area of a circle of radius R is calculated as:

A = pi*R^2

Now, if we have a sector of an angle θ degrees, the area of that sector is:

A = (θ/360°)*pi*R^2

In this case, we know that:

R = 15yd

And the area of the sector is 609 yd^2

Then we can replace these two values in the equation to get:

609yd^2  =(θ/360°)*3.14*(15yd)^2

(609yd^2)*360°/(3.14*(15yd)^2) = θ = 310.3°

And we want the angle also in radians.

We know that:

3.14 rad = 180°

(3.14 rad/180°) = 1

Then:

310.3° = 310.3°*(3.14 rad/180°) = (310.3°/180°)*3.14 rad = 5.41 radians.

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