A cable hangs between two poles of equal height and 28 feet apart. Set up a coordinate system where the poles are placed at x=−14 and x=14, where x is measured in feet. The height (in feet) of the cable at position x is:
h(x)=20 cosh(x/20)
where cosh(x)=(e^x+e^(-x))/2 is the hyperbolic cosine, which is an important function in physics and engineering. Find the length of the cable.
I know the arc length formula is L = int[ sqrt(1 + h'(x)^2)] from -14 to 14 and that's all I got.
While I have dealt with arc lengths and surface areas, all of my math skills go out the window with parabolics. Any help is greatly appreciated!
Calculate the arc length h(x)=20 cosh(x/20) from -14 to 14?
2012-02-27 2:26 am
回答 (2)
2012-02-27 2:41 am
✔ 最佳答案
Note that √(1 + (dy/dx)^2)= √(1 + (sinh(x/20))^2)
= √(1 + sinh^2 (x/20))
= √cosh^2 (x/20), since cosh^2(t) - sinh^2(t) = 1
= cosh(x/20).
Therefore, the arc length equals
∫(x = -14 to 14) cosh(x/20) dx
= 2 ∫(x = 0 to 14) cosh(x/20) dx, since the integrand is even
= 2 * 20 sinh(x/20) {for x = 0 to 14}
= 40 sinh(7/10).
I hope this helps!
2016-12-13 12:27 am
A cube has part lengths that are =, so it extremely is secure to assume that the two aspects is 14 cm. the equation to discover the quantity of a cube is V=s^3, so if s=14 cm, V=(14 cm)^3, it is 2744 cm^3
收錄日期: 2021-04-30 01:17:47
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