Question

perform the calculation below and round to the appropriate number of significant figures.\\(\frac{8.071}{4.216}\\)

Explanation:

Step1: Perform the division

We need to calculate the value of $\frac{8.071}{4.216}$. Using a calculator, we get:
$$\frac{8.071}{4.216} \approx 1.914373814$$

Step2: Determine significant figures

Both the numerator (8.071) and the denominator (4.216) have 4 significant figures. When dividing, the result should have the same number of significant figures as the least precise measurement (in terms of significant figures), which is 4 here. But let's check the intermediate result. Wait, actually, 8.071 has 4 significant figures, 4.216 has 4 significant figures. So the result should be rounded to 4 significant figures? Wait, no, wait: 8.071 is 4 sig figs, 4.216 is 4 sig figs. So the division result should be rounded to 4 significant figures? Wait, but let's see the calculation again. Wait, 8.071 ÷ 4.216. Let's do the division more accurately. 8.071 ÷ 4.216: 4.216 × 1.9 = 7.9904, 8.071 - 7.9904 = 0.0806. 0.0806 ÷ 4.216 ≈ 0.01912. So total is 1.9 + 0.01912 ≈ 1.91912? Wait, maybe my initial calculator was wrong. Let's use a calculator properly. 8.071 divided by 4.216:

8.071 ÷ 4.216 ≈ 1.914373814

Now, the number of significant figures: the numerator has 4 (8,0,7,1), denominator has 4 (4,2,1,6). So the result should be rounded to 4 significant figures? Wait, no, wait: when dividing, the rule is that the result has the same number of significant figures as the quantity with the least number of significant figures. Both have 4, so the result should have 4 significant figures. But let's check the first digit: 1.914373814. The first four significant figures are 1,9,1,4. The next digit is 3, which is less than 5, so we round down. Wait, but wait, maybe I made a mistake in the significant figures. Wait, 8.071 is 4 sig figs, 4.216 is 4 sig figs. So the result should be 1.914? Wait, no, wait, let's check the calculation again. Wait, 8.071 ÷ 4.216:

Let me use a calculator: 8.071 ÷ 4.216 = 1.914373814

Now, significant figures: 1.914373814. The first four significant figures are 1,9,1,4. The next digit is 3, so we keep the fourth digit as is. So 1.914? Wait, but wait, maybe the problem expects us to consider that both numbers have 4 significant figures, so the result should be rounded to 4 significant figures. But let's check the problem statement again. Wait, maybe I misread: 8.071 is 4 sig figs, 4.216 is 4 sig figs. So the division gives approximately 1.914, which when rounded to 4 significant figures is 1.914? Wait, no, 1.914373814: the first four significant figures are 1 (1st), 9 (2nd), 1 (3rd), 4 (4th). The next digit is 3, so we don't round up. So 1.914. But wait, maybe the problem is that 8.071 has 4 sig figs, 4.216 has 4 sig figs, so the result should have 4 sig figs. Alternatively, maybe the problem is expecting us to round to 4 significant figures, but let's check the calculation again. Wait, maybe I made a mistake in the division. Let me do it again: 4.216 × 1.914 = 4.216×1 + 4.216×0.9 + 4.216×0.014 = 4.216 + 3.7944 + 0.059024 = 8.069424, which is very close to 8.071. So 1.914 is correct. So the result is approximately 1.914 when rounded to 4 significant figures. Wait, but let's check the original numbers: 8.071 (4 sig figs), 4.216 (4 sig figs). So the answer should be 1.914? Wait, but maybe the problem is expecting 1.91, but no, 4 sig figs. Wait, maybe I messed up the significant figures. Wait, 8.071: the zero is between two non-zero digits, so it's significant. So 8,0,7,1: 4 sig figs. 4.216: 4,2,1,6: 4 sig figs. So division: result has 4 sig figs. So 1.914. But let's check with a calculator: 8.071 ÷ 4.216 = 1.914373814. So rounding to 4 significant figures: look at the 5th digit, which is 3, so we round down. So 1.914. But wait, maybe the problem is that 8.071 has 4 sig figs, 4.216 has 4 sig figs, so the answer is 1.914. Alternatively, maybe the problem is expecting 1.91, but no, 4 sig figs. Wait, maybe I made a mistake in the significant figures rule. Wait, the rule for multiplication and division is that the result has the same number of significant figures as the factor with the least number of significant figures. So if both have 4, the result has 4. So 1.914.

Answer:

$\boxed{1.914}$