use differentials to approximate the maximum resulting error in computing the area of a triangle?

Ryan G: use differentials to approximate the maximum resulting error in computing the area of a triangle?
The two legs of the triangle are measured to be a=500ft and b=700ft, with a possible error of 1 ft in each measurement. The angle between them is measured to be 30 degrees with an error of 0.25 degrees. The area is triangular so its area is given by A=1/2 (a)(b)sin(theta). Use differentials to approximate the maximum resulting error in computing the area of the triangle.

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Answer by bpiguy
In this particular problem, everything is positive, so the maximum error is simply the difference between your calculated area and the greatest possible value or the least possible value, whichever difference is greater.

The area formula is A = 1/2 (a +/- da) (b +/- db) sin (t +/- dt)

A = 1/2 (500 +/- 1) (700 +/- 1) sin (30 +/- 0.25)

The sine is an increasing function at theta = 30, so

A(max) = 1/2 x 501 x 701 sin 30.25 = 89123.990 = 89124.0
A(min) = 1/2 x 499 x 699 sin 29.75 = 86540.406 = 86540.4

Now, the area you’ll calculate is

A(calc) = 1/2 x 500 x 700 sin 30 = 350000 / 4 = 87500

Error 1 = A(max) – A(calc) = 1624 = 1.822% [use A(max) in the denominator to calculate percentage]

Error 2 = A(min) – A(calc) = -959.6 = 1.109% [use A(min) in the denominator to calculate percentage]

To get error percentages, you always want to divide by the actuals. Thus, if your estimate was 9, and the actual was 10, the error percentage is 1/10 or 10%, rather than 1/9 or 11.1%.

In your problem, the greatest error, both in absolute value and in percentage terms, is if your estimate is on the low side. There’s a substantial difference between 1624 and 960, and between 1.8% and 1.1%.

Also note that the “differentials” here didn’t mean much because everything was positive. If there were some negatives, we might have to expand binomials, differentiate, and set a derivative equal to zero to fins a max or min for the error term. Sometimes when you do that, you get expressions like da db which, when multiplied together, you can ignore, since that’s a small piece of a small piece.

Hope this is enough.

Answer by goober
dA = 1/2 (b) sin(theta) da = 350*0.5 * 1 = 175

dA = 1/2 (a) sin(theta) db = 250 *0.5 * 1 = 125

dA = 1/2 (a) (b) cos(theta) d(theta) = (1/2) * 500 * 700 * cos(30) * 0.25 * pi/180 = 661

max error = 175 + 125 + 661 = 961 sq ft