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Calculus/differential equations and propagating error

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Expert: - 10/31/2008


Question
The measurements of the base and altitude of a triangle are found to be 32 and
54 centimeters, respectively. The possible error in each measurement is 0.25
centimeter. Use differentials to approximate to one decimal place the possible
propagated error in computing the area of the triangle.  

Answer
The area of a triangle is bh/2 where b is the base and h is the height.

What we have is (32±0.25)(54±0.25).  
Multiply out the two and you get 1,728 ±8 ±13.5 ±0.0625, so a good way to go would be to say it is 1,728 ± 22.  
A good guess would probably be to round to the nearest 10's and get 1,730 ± 20.

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A point of trivia you might not care about.

It can be seen that 1,728 = 12^3 and 1^3 = 1.

Now 12^3 + 1^3 = 1,729, but 1,729 is also 10^3 + 9^3 since 10^3=1000 and 9^3=729.

1,729 is the smallest number with the property that it is the sum of cubes of two integers in two different ways.

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