Question

1. a balance accurate to one-hundredth of a gram measures the mass of a rock to be 56.10 grams. how many significant digits are in this value?

options: 2, 3, 4, there is not enough information to answer this question.

1. in the calculation shown below, how many significant digits should the answer have?

chart: 2.860 kilograms, 0.915 cubic meters, 1 cubic meter = 1.0×10⁶ cm³, 1000 grams = 1 kilogram
options: 1, 2, 3, 4

1. a star is measured to be 18.9 light years from earth. what is the correct answer in the calculation shown below? (consider both significant digits and scientific notation)

calculation: 18.9 light - years × (9.461×10¹² km / 1 light - year)
options: 1.8×10¹⁴, 1.79×10¹⁴, 17.9×10¹³, 17.88×10¹³

Explanation:

Problem 5:

Step1: Multiply the two numbers

We need to calculate \( 18.9 \times 9.461 \times 10^{12} \). First, multiply \( 18.9 \) and \( 9.461 \).
\( 18.9\times9.461 = 18.9\times(9 + 0.461)=18.9\times9+18.9\times0.461 = 170.1+8.7129 = 178.8129\)

Step2: Adjust the scientific notation

Now we have \( 178.8129\times 10^{12} \). To write this in proper scientific notation, we move the decimal point two places to the left, so we increase the exponent of 10 by 2.
\( 178.8129\times 10^{12}=1.788129\times 10^{14}\)

Step3: Consider significant figures

The number \( 18.9 \) has 3 significant figures and \( 9.461\times 10^{12} \) has 4 significant figures. When multiplying, the result should have the same number of significant figures as the number with the least number of significant figures, which is 3. So we round \( 1.788129\times 10^{14} \) to three significant figures.
\( 1.788129\times 10^{14}\approx1.79\times 10^{14}\)

To determine the number of significant digits in the result of a calculation, we look at the number of significant digits in each factor. The factors are \( 2.860 \) (4 significant digits), \( 0.915 \) (3 significant digits), \( 1.0\times 10^{6} \) (2 significant digits), and \( 1000 \) (assuming it's an exact conversion, so we don't consider its significant digits). When multiplying, the result should have the same number of significant digits as the factor with the least number of significant digits, which is \( 1.0\times 10^{6} \) with 2 significant digits? Wait, no, let's re - evaluate. Wait, \( 2.860 \) has 4, \( 0.915 \) has 3, \( 1.0\times 10^{6} \) has 2, but \( 1000\) grams to 1 kilogram is an exact conversion (defined), so we ignore it. The rule for multiplication/division is that the result has the same number of significant figures as the input with the fewest significant figures. Among \( 2.860 \) (4), \( 0.915 \) (3), \( 1.0\times 10^{6} \) (2), the least is 2? Wait, no, maybe I misread. Wait, the calculation is \( 2.860\times0.915\times1.0\times 10^{6}\times\frac{1}{1000} \) (since \( 1000\) grams = 1 kilogram, so dividing by 1000). Let's calculate the significant figures step by step. \( 2.860 \) (4 sig figs), \( 0.915 \) (3 sig figs), \( 1.0\times 10^{6} \) (2 sig figs), and \( 1000 \) is exact. When multiplying/dividing, the result should have the same number of sig figs as the number with the least sig figs. Here, \( 1.0\times 10^{6} \) has 2 sig figs? Wait, no, \( 1.0\times 10^{6} \) has two significant figures (the 1 and the 0). But \( 2.860 \) has 4, \( 0.915 \) has 3. Wait, maybe the question is about the number of significant digits in the final answer. Wait, the options are 1,2,3,4. Let's recalculate the significant figures. The number \( 2.860 \) has 4, \( 0.915 \) has 3, \( 1.0\times 10^{6} \) has 2, and \( 1000\) is exact. The rule is that when multiplying, the result's significant figures are determined by the least number of significant figures in the factors. So among 4,3,2, the least is 2? But wait, maybe I made a mistake. Wait, \( 1.0\times 10^{6} \) has two significant figures, \( 0.915 \) has three, \( 2.860 \) has four. So the answer should have 2 significant figures? But the options include 3. Wait, maybe the \( 1.0\times 10^{6} \) is considered to have two, but maybe the question is considering that \( 2.860 \) (4), \( 0.915 \) (3), and \( 1.0\times 10^{6} \) (2) - no, maybe the \( 1.0\times 10^{6} \) is a typo and should be \( 1.00\times 10^{6} \), but no. Wait, the correct approach: when multiplying, the number of significant figures in the result is equal to the number of significant figures in the least precise measurement. Here, \( 1.0\times 10^{6} \) has 2, \( 0.915 \) has 3, \( 2.860 \) has 4. So the answer should have 2? But the options are 1,2,3,4. Wait, maybe I misread the problem. Let's look again. The calculation is \( 2.860\) (maybe a volume) \( \times0.915\) (density) \( \times1.0\times 10^{6}\) (conversion) \( \times\frac{1}{1000}\) (conversion to kg). Wait, \( 2.860 \) has 4, \( 0.915 \) has 3, \( 1.0\times 10^{6} \) has 2, but \( 1000\) is exact. So the limiting factor is \( 1.0\times 10^{6} \) with 2? But maybe the \( 1.0\times 10^{6} \) is actually \( 1.00\times 10^{6} \), but no. Alternatively, maybe the question is about the number of significant digits in the product of \( 2.860\times0.915\times1.0\times 10^{6}\times\frac{1}{1000} \). Let's calculate the numerical value: \( 2.860\times0.915 = 2.6169\); \( 2.6169\times1.0\times 10^{6}=2.6169\times 10^{6}\); \( 2.6169\times 10…

Significant figures are the digits in a number that carry meaning contributing to its precision. For a number like \( 56.10 \) grams, all the digits are significant. The digits are 5, 6, 1, and 0. The trailing zero after the decimal is significant because it shows the precision of the measurement (accurate to one - hundredth of a gram). So we count all the digits: 5,6,1,0.

Answer:

\( 1.79\times 10^{14} \) (corresponding to the option \( 1.79\cdot10^{14} \))

Problem 4: