When a chain hangs under its own weight, its shape is a catenary (from the Latin word for chain). The equation of a catenary with a vertex on the y-axis is . Find the length of the chain from . Then use the same technique from Problem #8 to find the length of the parabola with the same vertex and endpoints.
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To find the length of the catenary and the parabola, we need to first determine their equations and then apply the appropriate formulas for finding arc lengths.
1. Finding the equation of the catenary:
A catenary is described by the equation y = a * cosh(x/a), where "a" is a constant that depends on the weight of the chain and how it is suspended. Since the catenary has a vertex on the y-axis, its equation becomes y = a * cosh(x/a) + a.
2. Length of the catenary:
The length of a curve can be found using the formula for arc length: L = ∫[a,b] √(1 + (dy/dx)^2) dx. For the catenary, we will use the equation y = a * cosh(x/a) + a, so we need to compute the derivative dy/dx.
Taking the derivative dy/dx of the catenary equation, we get dy/dx = sinh(x/a). Plugging this into the formula for arc length, we get:
L = ∫[a,b] √(1 + sinh^2(x/a)) dx.
Since we are given the endpoints (0, a) and (b, a), we can integrate from 0 to b:
L = ∫[0,b] √(1 + sinh^2(x/a)) dx.
To evaluate this integral, we can use trigonometric substitution or integration by parts.
3. Finding the equation of the parabola:
Since the vertex of the parabola is also on the y-axis and it has the same endpoints as the catenary, we can assume it is of the form y = ax^2 + a. To find the value of "a", we can use the fact that the parabola passes through the point (b, a).
Substituting (b, a) into the equation, we get:
a = a * b^2 + a.
Simplifying:
1 = b^2 + 1.
This implies that b^2 = 0, which in turn means that b = 0. Therefore, the equation of the parabola is y = ax^2 + a.
4. Length of the parabola:
To find the length of the parabola, we will again use the formula for arc length:
L = ∫[0,b] √(1 + (dy/dx)^2) dx.
Taking the derivative dy/dx of the parabola equation, we get dy/dx = 2ax. Plugging this into the formula for arc length, we have:
L = ∫[0,b] √(1 + (2ax)^2) dx.
Since b = 0, we have L = ∫[0,0] √(1 + (2ax)^2) dx = 0.
Therefore, the length of the parabola is zero.
In summary:
- The length of the catenary can be found by evaluating the integral L = ∫[0,b] √(1 + sinh^2(x/a)) dx.
- The length of the parabola is zero.