Catenary arch The portion of the curve y =17/15 - cosh x that ...

Question

Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

Accepted Answer

Hello. In this video we are going to let the curve y equal to 5/4 minus the coiking hyperbolic of x to define an arc above the x axis. What is the average height of the arc above the x axis? Now, in order to find the average height, we have an integral that's going to help us find that value. The average height is going to be defined as 1 divided by 2a multiplied by the integral from -a to a. Of F of X D X Now here, the function that is given to us is Y equal to 5/4 minus cosine hyperbolic of X. So that is going to be our function F of X for the integral. However, what we need to do is we need to go ahead and determine the value of our boundaries, which is also going to be in the denominator of the constant 1 divided by 2a. Now, because we know that the function given to us is defining an arc above the X axis, in order to integrate this function, we need to determine the X intercepts of the function. So to find the X intercepts, what we are going to do is take our function, set it equal to 0, and solve. Now, this is going to simplify to be cosine hyperbolic of x equal to 5/4, and in order to solve for x, we will take the inverse of cosine hyperbolic on both sides of the equation. So that will give us x is equal to cosine hyperbolic inverse of 5/4. And because the inverse is positive and negative, we will actually get two values for our boundary. So the boundary of A or the value of A we're going to be working with is A is equal to positive and negative cosine hyperbolic inverse of 5 divided by 4. So because this is a bit of a complicated value to work with, let's go ahead and first evaluate the integral before plugging in the value of a. We can rewrite the integral to be 1 divided by 2 multiplied by a multiplied by the integral from negative a to positive a of 5/4 minus cosine hyperbolic of xDX. No, The integral of cosine hyperbolic is going to be sine hyperbolic, so this is going to integrate to be 1 divided by 2a multiplied by 5/4 x minus sine hyperbolic of x, and this is going to be evaluated from negative A to A. Now, if we evaluate the the antiderivative at the boundaries, we get 1 divided by 2A multiplied by the quantity, 5/4 multiplied by positive A minus sine hyperbolic of A. -5/4 multiplied by a. Minus sine hyperbolic of negative A. Now here, there is one identity to keep in mind, and that identity is that sine hyperbolic of A is equivalent to negative sine hyperbolic of A. So this quantity here is going to simplify to be positive sine hyperbolic of A. And so by combining these terms together we can further simplify this integrand to be. 1 divided by 2 a. Multiplied by 5/4 a minus 2 sine hyperbolic of a. And by further combining like terms, we will have 5/4 minus sine hyperbolic of a divided by A. Now, if we were to plug in the value of A, which we determine to be cosine hyperbolic inverse of 5/4, then we will have 5/4. Minus sine hyperbolic. Of cosine hyperbolic inverse of a. Divided By cosine hyperbolic inverse of 5/4, which I made a small mistake here. This should not be a, this should be the constant of 5/4 in the numerator. Now, in order to simplify this numerator, there is one more identity we have to recall for sine hyperbolic, and it's the fact that sine hyperbolic of X is equivalent to the square root of cosine hyperbolic squared of X minus 1. So if we were to plug in cosine hyperbolic inverse into this function, then this is going to simplify to be 5/4 squared. -1 Which is going to further simplify to be 3 divided by 4. So that means our solution simplifies to be 5/4 minus 3 divided by 4, all divided by cosine hyperbolic inverse of 5/4. And by combining like terms together we can rewrite our solution to be 5 cosine hyperbolic inverse of 5/4 minus 3. Divided by the quantity 4 cosine hyperbolic inverse of 5/4. This is going to be the solution to the problem, with the overall solution being option C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

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