(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given y = 156 - (x - 40)^{2}/60. x = 0 and x = 85 Find distance traveled

2. Relevant equations

Arc Length S = integral of square root of ( y' )^{2}

3. The attempt at a solution

Doing this I get the trig sub tan(t) = (x - 40)/ 30 (Told by teacher to use this instead of -(x - 40)/30 )

so x = 30tan(t) + 40

and dx = 30sec^{2}(t)

So then under the radical I get 1 + ( - (tan (t) )^{2}

So is this equal to the square root of sec^{2}(t)?

Assuming it is, I get the integral of sec^{2}(t) times 30sec^{2}(t) or sec^{3}(t)

I know how to do this (I think) but I'm really not sure (I can't use what's written in the back of my book).

Doing the work I get:

S(x) = 15sec(t)tan(t) + (1/2) ln[ (sec(t) + tan(t) ]

Substitution x back in for t gives

tan(t) = (x-40)/30

and

sec(t) = square root [ 1 + ( ( -x + 40)/(30) )^{2}

But evaluating this from 0 to 85 gives me an arc length of ~ 75.05, which I know can't be right, but I can't figure out where I went wrong.

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# Calculating Distance Traveled (Arc Length)

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