You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters. If your measurement of the side is accurate to within 5%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary. A) Maximum error in volume is about ±29.4 cm³ B) Maximum error in volume is about ±411.6 cm³ C) Maximum error in volume is about ±2.1 cm³ D) Maximum error in volume is about ±5,762.4 cm³

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Quizplus · Accepted Answer

The maximum error in the measurement of the side of the cube is 5% of 14 cm, which is 0.7 cm. The error in volume calculation due to this error in measurement can be approximated by considering the derivative of the volume with respect to the side length, $V = s^3$, which gives $dV = 3s^2 ds$. Substituting $s = 14$ cm and $ds = 0.7$ cm, we get $dV = 3(14)^2(0.7) \approx 411.6$ cm³. This represents the maximum error in the volume calculation, which is about ±411.6 cm³.

You Measure the Side of a Cube to Be 14 143=2,74414 ^ { 3 } = 2,744143=2,744

You Measure the Side of a Cube to Be 14 $14 ^ { 3 } = 2,744$