determine the value of a such that the arc length of the catenary y = cosh(x) for -a ≤ x ≤ a equals 14…

determine the value of a such that the arc length of the catenary y = cosh(x) for -a ≤ x ≤ a equals 14. consider the shown work.\n\ns=int_{-a}^{a}sqrt{sinh^{2}(x)}dx=int_{-a}^{a}sinh(x)dx=cosh(x)\big|_{-a}^{a}=0\n\nthus, it is impossible for the arc length to equal 14.\n\nidentify the first error in the work shown.\nno errors exist in the work shown.\nthe square root was not needed.\nthe derivative of y was not calculated correctly.\nthe +1 under the square root in the arc length formula was missing.\nthe derivative y was not needed.

determine the value of a such that the arc length of the catenary y = cosh(x) for -a ≤ x ≤ a equals 14. consider the shown work.\n\ns=int_{-a}^{a}sqrt{sinh^{2}(x)}dx=int_{-a}^{a}sinh(x)dx=cosh(x)\big|_{-a}^{a}=0\n\nthus, it is impossible for the arc length to equal 14.\n\nidentify the first error in the work shown.\nno errors exist in the work shown.\nthe square root was not needed.\nthe derivative of y was not calculated correctly.\nthe +1 under the square root in the arc length formula was missing.\nthe derivative y was not needed.

Answer

Explanation:

Step1: Recuerda la fórmula de longitud de arco

La fórmula para la longitud de arco de una curva $y = f(x)$ en el intervalo $[x_1,x_2]$ es $s=\int_{x_1}^{x_2}\sqrt{1+(y')^{2}}dx$. Para $y = \cosh(x)$, la derivada $y'=\sinh(x)$. Entonces la longitud de arco en el intervalo $[-a,a]$ es $s=\int_{-a}^{a}\sqrt{1 + \sinh^{2}(x)}dx$. Pero en el trabajo mostrado, se olvidó el término $+ 1$ debajo de la raíz cuadrada, ya que se escribió $s=\int_{-a}^{a}\sqrt{\sinh^{2}(x)}dx$.

Answer:

The +1 under the square root in the arc length formula was missing.