A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 71.5 degree. How accurately must the angle be measured if the percent error in estimating the height of the tree must be less than 6%? Round the answer to the nearest hundredth of a degree.

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if the height of the tree is h, then if the angle is (x)

`h = 50tan(x)`

if the error in h is `Deltah` and the error corresponding in x is `Deltax`

`h+Deltah = 50tan(x+Deltax)`

if the percent error in h is 6%, `Deltah = 0.06h`

therefore,

`(h+Deltah)/h = (50tan(x+Deltax))/h`

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if the height of the tree is h, then if the angle is (x)

`h = 50tan(x)`

if the error in h is `Deltah` and the error corresponding in x is `Deltax`

`h+Deltah = 50tan(x+Deltax)`

if the percent error in h is 6%, `Deltah = 0.06h`

therefore,

`(h+Deltah)/h = (50tan(x+Deltax))/h`

`1.06 =(50tan(x+Deltax))/(50tan(x))`

but we know the angle is x = 71.5

therefore,

`tan(71.5+Deltax) = 1.06tan(71.5)`

`tan(71.5+Deltax) = 3.16800606`

`71.5+Deltax = 72.48`

`Deltax = 0.98`

`(Deltax)/(x) * 100 = 0.98/71.5 * 100 `

The percentage error that can happen in angle is = 1.37%

 

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