if the perimeter of a circular sector is fixed at 100 ft, what values of r and s give the sector the greatest area? ...?

Answers

Answer 1

Answer: The sector arc length is just a fraction of the circumference:
Laʀc = (2πR) • (θ ⁄ 2π) = R • θ ... where θ = central angle (of sector)

Ps = R + R + (R • θ) ... perimeter of sector = Ps

Ps = R • (2 + θ) ... Ps = 100 ft

100 = R • (2 + θ)

θ = (100 ⁄ R) – 2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...

Similarly, the sector area is just a fraction of the circle area:

As = (πR²) • (θ ⁄ 2π) ... area of sector = As

As = R² • θ ⁄ 2 ... substitute for θ

As = R² • [(100 ⁄ R) – 2 ] ⁄ 2

As = 50R – R² ... differentiate

As' = 50 – 2R ... set to zero

0 = 50 – 2R

R = 25 ft ... optimum radius

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...

As = 50R – R² ... evaluate at: R = 25 ft

As = 50(25) – (25)²

As = 625 ft² ... maximum area

Note ... at any other R_value, the sector area is less.

" θ " can be determined using: θ = (100 ⁄ R) – 2

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Answer 2

Answer:

Final answer:

For a circular sector with a fixed perimeter of 100 ft, the values of radius (r) and arc length (s) that will maximize the sector's area are r=25 and s=50.

Explanation:

The perimeter of a circular sector is composed by the length of the arc (s) plus twice the radius (r). If this sum is fixed at 100 ft, then the length of the arc s is equal to 100 - 2r. The area A of a circular sector can be defined as A = 0.5 * r * s.

Substituting the expression for s into the area formula obtains A = 0.5 * r * (100-2r). Simplifying results in A = 50r - r^2 which is a downward opening parabola.

The maximum value of a parabola occurs at the vertex. For a parabola in the form y=ax^2 + bx + c, the x-coordinate of the vertex is -b/(2a). In this case, a=-1 and b=50, hence r=-50/2*(-1) = 25. Substituting r=25 back into the formula for s obtains s = 100-2*25 = 50. Therefore, the values for r and s that will give the circular sector the greatest area are r=25 and s=50.

Learn more about Perimeters and Areas here:

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Answers

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we get:
30*x - 25*x = 90000
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Answers

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Answers

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Answers

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Answers

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I hope that's help ! if you have question please ask

2/5-1/4

2/5=8/20

1/4=5/20

8/20-5/20=3/20
The answer to the problem is3/20

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