A description of the Gateway Arch is y=1260 - 315(e^0.00418x + e^-0.00418x), where the base of the arch is [-315, 315] and x and y are measured in ft. Find the average height of the arch above the...

A description of the Gateway Arch is

y=1260 - 315(e^0.00418x + e^-0.00418x),

where the base of the arch is [-315, 315] and x and y are measured in ft.

Find the average height of the arch above the ground.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The height of the arch is equal to the extreme value of function, hence, you need to evaluate the critical value solving the equation `y' = 0,` such that:

`y' = (1260 - 315(e^(0.00418x) + e^(-0.00418x)))'`

`y' = -315*0.00418e^(0.00418x) + 315*0.00418e^(-0.00418x)`

`y' = 0 => -315*0.00418e^(0.00418x) + 315*0.00418e^(-0.00418x) = 0`

Factoring out `-315*0.00418` yields:

`-315*0.00418*(e^(0.00418x) - e^(-0.00418x)) = 0 => e^(0.00418x) - e^(-0.00418x) = 0`

You need to perform the substitution `e^(0.00418x) = t` , such that:

`t - t^(-1) = 0 => t - 1/t = 0 => t^2 - 1 = 0 => t^2=1 => t_(1,2) = +-1`

`e^(0.00418x) = t_1 => e^(0.00418x) = 1 => ln (e^(0.00418x)) = ln 1 => 0.00418x = 0 => x = 0`

`e^(0.00418x) = t_2 => e^(0.00418x) = -1` invalid since `e^(0.00418x) > 0`

Hence, the function reaches its extreme value at `x = 0` , such that:

`y|_(x = 0) = 1260 - 315(e^(0) + e^(0)) => y|_(x = 0) = 1260 - 315*2 => y|_(x = 0) = 630`

Hence, evaluating the average height of the gateway arch whose equation is given by `y = 1260 - 315(e^(0.00418x) + e^(-0.00418x))` yields `y|_(x = 0) = 630` ft.

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