Question

Perform the calculations and determine the absolute and percent relative uncertainty. Express each answer with the correct number of significant figures.
a. [9.8(±0.3)−2.31(±0.01)]8.5(±0.6)= __________
b. absolute uncertainty: __________
c. absolute uncertainty: _________
d. percent relative uncertainty: ___________

Answer 1

Answer:

Explanation:

Given the equation:

$\implies \dfrac{[9.8(\pm0.3)-2.31(\pm 0.01)]}{8.5(\pm0.6)}$

The absolute uncertainty in a measurement is the term used to describe the degree of inaccuracy.

The first step is to determine the algebraic value on the numerator.

Algebraic value = 9.8 - 231

= 7.49

The absolute uncertainty = $\sqrt{(abs. uncertainty_{v_1})^2+(abs. uncertainty_{v_2})^2}$

absolute uncertainty = $\sqrt{(0.3)^3 + (0.01)^2}$

= $\sqrt{0.09 + 0.0001}$

= 0.300167

∴

[9.8(±0.3) - 2.31(±0.01)] = 7.49(±0.300167)

The division process now is:

$\implies \dfrac{[9.8(\pm0.3)-2.31(\pm 0.01)]}{8.5(\pm0.6)}= \dfrac{7.49 (\pm 0.300167)}{8.5 (\pm0.6)}$

Relative uncertainty = $\dfrac{(\pm 0.300167)}{7.49}\times 100 \ , \ \dfrac{(\pm 0.6) }{8.5} \times 100$

Relative uncertainty = ±4.007565% ,  ±7.058824%

$\text{Relative uncertainty} = \sqrt{(4.007565)^2+(7.058824)^2}$

$\text{Relative uncertainty} = \sqrt{16.06057723+49.82699626}$

$\text{Relative uncertainty} = \sqrt{65.88757349}$

$\text{Relative uncertainty} = 8.117116$

≅ 8%

The algebraic value = $\dfrac{7.49}{8.5}$

= 0.881176

≅ 0.88

The percentage of the relative uncertainty =$\dfrac{\text{Absolute uncertainty }}{\text{calculated value} }\times 100$

By cross multiplying:

$\text{Absolute uncertainty} (\%) = \dfrac{\text{relative uncertainty} \times \text{calculated value}}{100}$

$\text{Absolute uncertainty} (\%) = \dfrac{8.117116\times 0.881176}{100}$

$\text{Absolute uncertainty} (\%) = 0.0715260$

$\mathbf{\text{Absolute uncertainty} (\%) \simeq 0.07}$

Finally:

$\mathbf{\implies \dfrac{[9.8(\pm0.3)-2.31(\pm 0.01)]}{8.5(\pm0.6)}= 0.88 \pm (0.07) \pm 8\%}$