## Related questions with answers

The Gateway Arch in St. Louis was constructed using the equation

$y = 211.49 - 20.96 cosh 0.03291765x$

for the central curve of the arch, where x and y are measured in meters and |x| ≤ 91.20. Set up an integral for the length of the arch and use your calculator to estimate the length correct to the nearest meter.

Solutions

VerifiedArc Length Formula

If $f'(x)$ is continuous on $[a, b]$, then the length of the curve $y=f(x)$, $a \leq x \leq b$, is

$L =\int_a^b \sqrt{1+\left[f'(x) \right]^2}\hspace{1mm}dx$

The equation for the central curve of the arch is given as

$y=211.49-20.96\cosh0.03291765x$

Therefore

$f'(x)=\dfrac{d\left(211.49-20.96\cosh0.03291765x\right)}{dx}$

$\begin{align*} y&=211.49 -20.96 \cosh(0.03291765x)\ \ \text{for}\ \ \left|x \right| \leq 91.20\\ \dfrac{dy}{dx}& = -20.96 \times( 0.03291765) \sinh(0.03291765x)\\ \end{align*}$

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