Question

1. the sides of a small rectangular box are measured to be 1.80 +/- 0.01 cm long, 2.05 +/- 0.02 cm long, and 3.10 +/- 0.01 cm long. calculate the box’s volume and approximate uncertainty in cubic centimeters. (1 point) 11 +/- 0.21 cm^3 11.4 +/- 0.21 cm^3 11.4 +/- 0.2 cm^3 11 +/- 0.2 cm^3

Explanation:

Step1: Calculate the volume

The volume $V$ of a rectangular - box is given by $V = l\times w\times h$, where $l = 1.80$ cm, $w = 2.05$ cm, and $h = 3.10$ cm.
$V=1.80\times2.05\times3.10 = 11.397\approx11.4$ $cm^{3}$

Step2: Calculate the relative uncertainty

The formula for relative uncertainty in a product $y = a\times b\times c$ is $\frac{\Delta y}{y}=\sqrt{(\frac{\Delta a}{a})^{2}+(\frac{\Delta b}{b})^{2}+(\frac{\Delta c}{c})^{2}}$.
Here, $a = 1.80$ cm, $\Delta a=0.01$ cm; $b = 2.05$ cm, $\Delta b = 0.02$ cm; $c = 3.10$ cm, $\Delta c=0.01$ cm.
$\frac{\Delta a}{a}=\frac{0.01}{1.80}\approx0.0056$, $\frac{\Delta b}{b}=\frac{0.02}{2.05}\approx0.0098$, $\frac{\Delta c}{c}=\frac{0.01}{3.10}\approx0.0032$.
$\frac{\Delta V}{V}=\sqrt{(0.0056)^{2}+(0.0098)^{2}+(0.0032)^{2}}=\sqrt{0.00003136 + 0.00009604+0.00001024}=\sqrt{0.00013764}\approx0.0117$.

Step3: Calculate the absolute uncertainty

$\Delta V=V\times\frac{\Delta V}{V}=11.397\times0.0117\approx0.133\approx0.2$ $cm^{3}$

Answer:

$11.4\pm0.2$ $cm^{3}$