Question

multiply or divide the following measurements. be sure each answer you enter contains the correct number of significant digits.
$362.6\\ \text{m} \div 79.075\\ \text{s} = \square\\ \dfrac{\text{m}}{\text{s}}$
$20.947\\ \dfrac{\text{mol}}{\text{l}} \times 50.\text{ l} = \square\\ \text{mol}$
$20.947\\ \dfrac{\text{g}}{\text{ml}} \times 36.\text{ ml} = \square\\ \text{g}$

Explanation:

First Calculation: \( 362.6\ \text{m} \div 79.075\ \text{s} \)

Step1: Perform the division

\( 362.6 \div 79.075 \approx 4.585 \)

Step2: Determine significant digits

The least number of decimal places or significant digits? Wait, 362.6 has 4 significant digits, 79.075 has 5. When dividing, the result should have the same number of significant digits as the least precise measurement. 362.6 has 4, so we round to 4 significant digits? Wait, no: 362.6 is 4 sig figs, 79.075 is 5. So the result should have 4 sig figs? Wait, 362.6 ÷ 79.075 ≈ 4.585. Wait, 362.6 has 4, 79.075 has 5. So the answer should have 4 significant digits? Wait, no, the rule is that for multiplication/division, the result has the same number of significant digits as the input with the least number of significant digits. 362.6 has 4, 79.075 has 5. So 4 sig figs. Wait, 362.6 is 4, 79.075 is 5. So 4.585 rounded to 4 sig figs is 4.585? Wait, no, 4.585 is 4 sig figs? Wait, 4.585: the first four digits are 4,5,8,5. Wait, maybe I miscalculated. Let's do the division again: 362.6 ÷ 79.075. Let's compute that: 362.6 ÷ 79.075 ≈ 4.585 (using calculator). So with 4 sig figs, it's 4.585? Wait, 362.6 has 4, so the answer should have 4. So 4.585 (or maybe 4.585, but let's check: 362.6 is 4, 79.075 is 5. So 4 sig figs. So 4.585 (which is 4 sig figs? Wait, 4.585 has 4? Wait, 4.585: digits are 4,5,8,5 – four significant digits. So that's correct.

Second Calculation: \( \frac{20.947\ \text{mol}}{\text{L}} \times 50.\ \text{L} \)

Step1: Multiply the numbers

\( 20.947 \times 50. = 1047.35 \)

Step2: Determine significant digits

1. has 2 significant digits (the decimal shows that the zero is significant), 20.947 has 5. So the result should have 2 significant digits? Wait, no: 50. is two sig figs (the decimal makes the zero significant), 20.947 is five. So when multiplying, the result should have two sig figs? Wait, 20.947 × 50. = 1047.35. Rounded to two sig figs: 1.0 × 10³? Wait, no, 50. is two sig figs, so 20.947 × 50. = 1047.35 ≈ 1.0 × 10³? Wait, no, 50. is two sig figs, so 20.947 × 50. = 1047.35. Let's see: 20.947 (5 sig figs) × 50. (2 sig figs) = result with 2 sig figs. So 1.0 × 10³? Wait, no, 20.947 × 50. = 1047.35. 50. is two sig figs, so 1.0 × 10³? Wait, 1047.35 rounded to two sig figs is 1.0 × 10³ (which is 1000 with two sig figs). Alternatively, maybe 50. is considered two sig figs, so 20.947 × 50. = 1047.35 ≈ 1.0 × 10³ (or 1000 with two sig figs). Wait, but maybe 50. is two sig figs, so 20.947 × 50. = 1047.35 ≈ 1.0 × 10³ (or 1000, but written as 1.0 × 10³ to show two sig figs). Wait, but maybe I made a mistake. Wait, 50. has two sig figs, 20.947 has five. So the product should have two sig figs. So 20.947 × 50. = 1047.35 ≈ 1.0 × 10³ (or 1000 with two sig figs). Alternatively, maybe 50. is two sig figs, so 1.0 × 10³ mol.

Third Calculation: \( \frac{20.947\ \text{g}}{\text{mL}} \times 36.\ \text{mL} \)

Step1: Multiply the numbers

\( 20.947 \times 36. = 754.092 \)

Step2: Determine significant digits

1. has 2 significant digits (the decimal shows the zero? Wait, 36. is two sig figs (the decimal indicates that the 6 is significant, and the 3 is too, so two sig figs). 20.947 has 5. So the result should have two sig figs. So 754.092 ≈ 7.5 × 10² (or 750 with two sig figs, written as 7.5 × 10²).

Answer:

First: \( \boldsymbol{4.59}\ \text{m/s} \) (Wait, maybe I messed up the first one. Let's recalculate 362.6 ÷ 79.075. Let's do it more accurately: 362.6 ÷ 79.075. Let's compute 79.075 × 4.58 = 79.075 × 4 + 79.075 × 0.58 = 316.3 + 45.8635 = 362.1635. Which is close to 362.6. So 4.58 + (362.6 - 362.1635)/79.075 ≈ 4.58 + 0.4365/79.075 ≈ 4.58 + 0.0055 ≈ 4.5855. So approximately 4.59 (three sig figs? Wait, 362.6 has four, 79.075 has five. So maybe the answer should have four sig figs: 4.586? Wait, 362.6 is four, 79.075 is five. So four sig figs: 4.586.

Second: \( \boldsymbol{1.0 \times 10^3}\ \text{mol} \) (or 1000 mol with two sig figs, but 20.947 × 50. = 1047.35, which with two sig figs is 1.0 × 10³)

Third: \( \boldsymbol{7.5 \times 10^2}\ \text{g} \) (or 750 g with two sig figs, since 36. has two sig figs)

Wait, maybe I made a mistake in significant digits. Let's recheck:

1. \( 362.6\ \text{m} \div 79.075\ \text{s} \): 362.6 has 4 sig figs, 79.075 has 5. So the result should have 4 sig figs. 362.6 ÷ 79.075 ≈ 4.585, which is 4 sig figs (4.585).

1. \( \frac{20.947\ \text{mol}}{\text{L}} \times 50.\ \text{L} \): 50. has 2 sig figs, 20.947 has 5. So result has 2 sig figs. 20.947 × 50. = 1047.35 ≈ 1.0 × 10³ (2 sig figs).

1. \( \frac{20.947\ \text{g}}{\text{mL}} \times 36.\ \text{mL} \): 36. has 2 sig figs, 20.947 has 5. So result has 2 sig figs. 20.947 × 36. = 754.092 ≈ 7.5 × 10² (2 sig figs).

So final answers:

1. \( \boldsymbol{4.585}\ \text{m/s} \) (or 4.59 if rounded to three, but 4 sig figs is 4.585)

1. \( \boldsymbol{1.0 \times 10^3}\ \text{mol} \)

1. \( \boldsymbol{7.5 \times 10^2}\ \text{g} \)