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4 years ago

How do you these questions?

Mathematics

1 answer:

andrezito [222]4 years ago

7 0

Answer:

a) s = 2a sinh(b/a)

b) 18 feet

Step-by-step explanation:

The arc length is:

s = ∫ √(1 + (dy/dx)²) dx

y = a cosh(x/a)

y = ½ a (e^(x/a) + e^(-x/a))

dy/dx = ½ a (1/a e^(x/a) − 1/a e^(-x/a))

dy/dx = ½ (e^(x/a) − e^(-x/a))

(dy/dx)² = ¼ (e^(2x/a) − 2 + e^(-2x/a))

Plugging in:

s = ∫ √(1 + ¼ (e^(2x/a) − 2 + e^(-2x/a))) dx

s = ∫ ½ √(4 + e^(2x/a) − 2 + e^(-2x/a)) dx

s = ∫ ½ √(e^(2x/a) + 2 + e^(-2x/a)) dx

s = ∫ ½ √(e^(x/a) + e^(-x/a))² dx

s = ∫ ½ (e^(x/a) + e^(-x/a)) dx

s = ½ (a e^(x/a) − a e^(-x/a))

s = ½ a (e^(x/a) − e^(-x/a))

Evaluate from x=-b to x=b:

s = ½ a (e^(b/a) − e^(-b/a)) − ½ a (e^(-b/a) − e^(b/a))

s = ½ a e^(b/a) − ½ a e^(-b/a) − ½ a e^(-b/a) + ½ a e^(b/a)

s = a e^(b/a) − a e^(-b/a)

s = a (e^(b/a) − e^(-b/a))

s = 2a sinh(b/a)

We know that s = 48, and b = 20. We also know y = 30 when x = b.

48 = 2a sinh(20/a)

24 = a sinh(20/a)

30 = a cosh(20/a)

Using the given identity:

cosh²(20/a) − sin²(20/a) = 1

a² cosh²(20/a) − a² sin²(20/a) = a²

30² − 24² = a²

a² = 324

a = 18

Therefore, the wire will be 18 feet above the ground at the lowest point.

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