Respuesta :
Answer:
Area of another sector=[tex]53.0292 in^2[/tex]
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=[tex]\frac{arc\;length} {Circumference\;of\;circle}\times \pi r^2[/tex]
Area of circular sector=[tex]\frac{l}{2\pi r}\times \pi r^2=\frac{l}{2}\times r[/tex]
Where Circumference of circle=[tex]2\pi r[/tex]
Substitute the values
Area of circular sector=[tex]\frac{12.84}{2}\times (8.26}[/tex]
Area of circular sector=[tex]53.0292 in^2[/tex]
Area of another sector=[tex]53.0292 in^2[/tex]
Perimeter of circular sector=8.26+8.26+12.84
Perimeter of circular sector=29.36 in
Perimeter of another sector=29.36 in
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;
- The radius of an alternative sector is 6.42 inches
- The arc length of an alternative sector is 16.52 inches
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
- [tex]\displaystyle Area \ of \ sector = \frac{12.84}{16.52 \cdot \pi} \times 68.2276 \cdot \pi = 53.0292[/tex]
Area of the sector, A = 53.0292 inch²
Therefore, we have;
2·R + A = 29.36 inches
[tex]\displaystyle \mathbf{\frac{A}{2\cdot \pi \cdot R} \times \pi \cdot R^2} = 53.0292[/tex]
A·R = 53.0292 × 2
[tex]\displaystyle New \ arc \ length, \ A = \mathbf{\frac{53.0292 \times 2}{R}}[/tex]
Which gives;
[tex]\displaystyle 2 \cdot R + \frac{53.0292 \times 2}{R} = 29.36[/tex]
[tex]\displaystyle R + \frac{53.0292 }{R} = 14.68[/tex]
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 radius of the other sector, R = 6.42 inches
The arc length, A = 29.36 - 2·R
∴ A = 29.36 - 2 × 6.42 = 16.52
- The arc length of the other sector, A = 16.52 inches
Learn more about the sector of a circle here:
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