A surveying team set up some equipment 121 ft from the base...

A surveying team set up some equipment 121 ft from the base of a tree in order to sight the top of the tree. From ground level, they measure the angle of elevation to be 31°. If the calculated height needs to be accurate to within 2%, what is the allowed error in the angle measurement? (Assume the 121 ft measurement is 100% accurate.) The allowed relative error is .0061 . The allowed percent error is .01968 %. Note: enter an answer accurate to 4 decimal places.

Transcript

00:04 So in this question, basically what we have is that you have a tree, right? suppose this tree somewhere, there's something like this.

00:11 Okay, this is the tree we look at.

00:14 And you have equipment, you want to measure the height of the tree, right? so the equipment is somewhere here, you put it here, right? and the distance of this equipment to the base of the tree, according to the question, is 221 feet.

00:26 And then you basically, next step, what you do is to measure the angle, right? ceta, right? and you measure the angle, something is about 31 degrees, right? and basically, you want to find out the heart of the tree, right? and so, you know, the head of a tree, obviously, is going to be given by 121, right, times tangent of ceta, right? and of course, the measurement of ceta can be, can have some uncertainty.

00:54 As a result, the tree will also have some uncertainty, right? you can see the uncertainty, which i'm going to call it, delta h, and that's what's given by, by doing a definition, right, given by, you know, the uncertainty in this angle theta and divided by, of course, cosine cedar squared, right? it's just the, the different, the derivatives of tangent theta, right? and of course, the, this can be for the, uh, written as 122 over, you know, cosine cossin squared ceta is 31, 31 degrees, right, and times data cata, right? and of course, this can be, you can work out this, right? and this is obviously roughly given by 121 times.

01:39 This is obviously is going to be in a consensus degree, it's actually, you know, 3 over 4, right? so this square is 3 over 4, so actually 4 over 3 basically, right? and times data cata, right? now the data hedge according to the question must be, needs to be actually accurate within 2%.

02:00 So in other words, data h, if we divide this both sides by, if we decide both sides by h, then data h over h, the maximum should be about 2%...

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