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Answer:
The maximum error in the calculated surface area is approximately 8.3083 square centimeters.
Step-by-step explanation:
The circumference (), in centimeters, and the surface area (
), in square centimeters, of a sphere are represented by following formulas:
(1)
(2)
Where is the radius of the sphere, in centimeters.
By applying (2) in (1), we derive this expression:
(3)
By definition of Total Differential, which is equivalent to definition of Linear Approximation in this case, we determine an expression for the maximum error in the calculated surface area (), in square centimeters:
(4)
Where:
- Measure circumference, in centimeters.
- Possible error in circumference, in centimeters.
If we know that and
, then the maximum error is:
The maximum error in the calculated surface area is approximately 8.3083 square centimeters.
Absolute value is a function that takes some number and outputs the size of that number, regardless of the sign. By definition, if
, and
if
.
For example,
- |2| = 2 because the input 2 is positive, so the input is unchanged;
- |-3| = -(-3) = 3 because the input -3 is negative, so the sign is switched.
What this means for this problem is that there are two cases to consider:
- If
, then
, so that
- On the other hand, if
, then
, so