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a homeowner has 200 feet of fence to enclose an area for a pet.

(a) if the area is given by A(x)=X(100-x), what dimension maximize the area inside the fence?

(b) what is the maximum area?

(c) determine the domain and range of A in this application.

Respuesta :

sqdancefan

Answer:

  • (a) x = 50 maximizes the area. We assume that means 50 ft × 50 ft
  • (b) 2500 ft²
  • (c) domain: 0 ≤ x ≤ 100; range: 0 ≤ A ≤ 2500

Step-by-step explanation:

(a) The function describes a parabola that opens downward. The value of A(x) is zero when x=0 and when x=100. The vertex (maximum) is halfway between those zeros, at x=50. The dimensions are 50 and 100-50 = 50. The maximum area pen will be 50 ft square.

(b) A(50) = (50 ft)² = 2500 ft² . . . the maximum area

(c) A(x) will be negative if x < 0 or x > 100, so the domain is 0 ≤ x ≤ 100.

The value of A(x) ranges from 0 to its maximum, 2500. Hence the range is 0 ≤ A(x) ≤ 2500.

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