Question

Calculate the following to the correct number of significant figures. Assume that all these numbers are measurements. (a) \(x=17.2+65.18-2.4\) (b) \(x=\frac{13.0217}{17.10}\) (c) \(x=(0.0061020)(2.0092)(1200.00)\) (d) \(x=0.0034+\frac{\sqrt{(0.0034)^{2}+4(1.000)\left(6.3 \times 10^{-4}\right)}}{(2)(1.000)}\) (e) \(x=\frac{\left(2.998 \times 10^{8}\right)\left(3.1 \times 10^{-7}\right)}{6.022 \times 10^{23}}\)

Short answer

Question: Calculate each expression and round the result to the correct number of significant figures. (a) \(17.2 + 65.18 - 2.4\) (b) \(\frac{13.0217}{17.10}\) (c) \((0.0061020)(2.0092)(1200.00)\) (d) \(0.0034+\frac{\sqrt{(0.0034)^2+4(1.000)(6.3\times10^{-4})}}{(2)(1.000)}\) (e) \(\frac{(2.998\times10^8)(3.1\times10^{-7})}{6.022\times10^{23}}\) Answers: (a) \(80.0\) (b) \(0.7606\) (c) \(14.67\) (d) \(0.029\) (e) \(1.543 \times 10^{-22}\)

Step-by-step solution

(a) Calculate sum and do subtraction

To calculate \(x = 17.2 + 65.18 - 2.4\), first, we need to perform the summation and subtraction: \(x = (17.2 + 65.18) - 2.4 = 82.38 - 2.4\) Now, subtract: \(x = 79.98\)

(a) Determine the correct number of significant figures

In the given expression, the least number of decimal places is 1 (for 17.2 and 2.4). Therefore, the answer should have 1 decimal place: \(x = 80.0\)

(b) Perform division

To calculate \(x =\frac{13.0217}{17.10}\), divide the numerator by the denominator: \(x = 0.760637 \dots\)

(b) Determine the correct number of significant figures

In the given expression, the least number of significant figures is 4 (for 17.10). Therefore, the answer should have 4 significant figures: \(x = 0.7606\)

(c) Perform multiplication

To calculate \(x = (0.0061020)(2.0092)(1200.00)\), multiply the numbers: \(x = 14.673611424 \dots\)

(c) Determine the correct number of significant figures

In the given expression, the least number of significant figures is 4 (for 0.0061020 and 2.0092). Therefore, the answer should have 4 significant figures: \(x = 14.67\)

(d) Calculate the expression

First, calculate the expression inside the square root: \(x=0.0034+\frac{\sqrt{(0.0034)^2+4(1.000)(6.3\times10^{-4})}}{(2)(1.000)}\) \(\Rightarrow x= 0.0034 + \frac{\sqrt{0.00001156+0.00252}}{2}\) \(\Rightarrow x= 0.0034 + \frac{\sqrt{0.00253156}}{2}\) Now, calculate the square root and divide by 2: \(x = 0.0034 + 0.02515 \dots\) Finally, add the two numbers: \(x = 0.02855 \dots\)

(d) Determine the correct number of significant figures

In the given expression, the least number of significant figures is 2 (for 0.0034). Therefore, the answer should have 2 significant figures: \(x = 0.029\)

(e) Perform multiplication and division

First, perform the multiplication: \(x=\frac{(2.998\times10^8)(3.1\times10^{-7})}{6.022\times10^{23}}\) \(\Rightarrow x= \frac{9.2958\times10^1}{6.022\times10^{23}}\) Now, perform the division: \(x = 1.543403 \dots \times 10^{-22}\)

(e) Determine the correct number of significant figures

In the given expression, the least number of significant figures is 4 (for 2.998 and 6.022). Therefore, the answer should have 4 significant figures: \(x = 1.543 \times 10^{-22}\)