If a raindrop weighs 160 mg (milligram) on average and \( 2.6 \times 10^{3} \) raindrops fall on the lawn every 10 minutes, what mass in kilogram \( (\mathrm{kg} \) ) of rain falls on the lawn in 6 mins? (Answer to I decimal place)

To find the mass of rain that falls on the lawn in 6 minutes, we can follow these steps: 1. **Calculate the number of raindrops that fall in 6 minutes.** - We know that \( 2.6 \times 10^{3} \) raindrops fall every 10 minutes. - Therefore, in 1 minute, the number of raindrops that fall is: \[ \text{Raindrops per minute} = \frac{2.6 \times 10^{3}}{10} \] - In 6 minutes, the total number of raindrops is: \[ \text{Total raindrops in 6 minutes} = \text{Raindrops per minute} \times 6 \] 2. **Calculate the total mass of the raindrops in milligrams.** - Each raindrop weighs 160 mg, so the total mass in milligrams is: \[ \text{Total mass (mg)} = \text{Total raindrops in 6 minutes} \times 160 \] 3. **Convert the total mass from milligrams to kilograms.** - Since \( 1 \, \text{kg} = 10^{6} \, \text{mg} \), we convert the mass to kilograms by dividing by \( 10^{6} \). Now, let's perform the calculations step by step. ### Step 1: Calculate the number of raindrops in 6 minutes \[ \text{Raindrops per minute} = \frac{2.6 \times 10^{3}}{10} = 2.6 \times 10^{2} \] \[ \text{Total raindrops in 6 minutes} = 2.6 \times 10^{2} \times 6 \] ### Step 2: Calculate the total mass in milligrams \[ \text{Total mass (mg)} = \text{Total raindrops in 6 minutes} \times 160 \] ### Step 3: Convert to kilograms \[ \text{Total mass (kg)} = \frac{\text{Total mass (mg)}}{10^{6}} \] Now, let's calculate these values. Calculate the value by following steps: - step0: Calculate: \(\frac{2.6\times 10^{3}}{10}\times 6\) - step1: Reduce the fraction: \(2.6\times 10^{2}\times 6\) - step2: Rewrite the expression: \(26\times 10\times 6\) - step3: Multiply the terms: \(260\times 6\) - step4: Multiply the numbers: \(1560\) Calculate or simplify the expression \( 160 * ((2.6 * 10^3) / 10 * 6) \). Calculate the value by following steps: - step0: Calculate: \(160\left(\frac{2.6\times 10^{3}}{10}\times 6\right)\) - step1: Remove the parentheses: \(160\times \frac{2.6\times 10^{3}}{10}\times 6\) - step2: Reduce the fraction: \(160\times 2.6\times 10^{2}\times 6\) - step3: Multiply the terms: \(2496\times 10^{2}\) - step4: Evaluate the power: \(2496\times 100\) - step5: Multiply: \(249600\) Calculate or simplify the expression \( 249600 / 10^6 \). Calculate the value by following steps: - step0: Calculate: \(\frac{249600}{10^{6}}\) - step1: Rewrite the expression: \(\frac{100\times 2496}{10^{6}}\) - step2: Rewrite the expression: \(\frac{10^{2}\times 2496}{10^{6}}\) - step3: Reduce the fraction: \(\frac{2496}{10^{4}}\) - step4: Rewrite the expression: \(\frac{64\times 39}{10^{4}}\) - step5: Rewrite the expression: \(\frac{64\times 39}{2^{4}\times 5^{4}}\) - step6: Rewrite the expression: \(\frac{2^{6}\times 39}{2^{4}\times 5^{4}}\) - step7: Reduce the fraction: \(\frac{2^{2}\times 39}{5^{4}}\) - step8: Calculate: \(\frac{156}{5^{4}}\) - step9: Evaluate the power: \(\frac{156}{625}\) Let's summarize the calculations: 1. **Total raindrops in 6 minutes:** \[ \text{Total raindrops in 6 minutes} = 1560 \] 2. **Total mass in milligrams:** \[ \text{Total mass (mg)} = 160 \times 1560 = 249600 \, \text{mg} \] 3. **Convert to kilograms:** \[ \text{Total mass (kg)} = \frac{249600}{10^{6}} = 0.2496 \, \text{kg} \] Finally, rounding to one decimal place, the mass of rain that falls on the lawn in 6 minutes is: \[ \text{Mass} \approx 0.2 \, \text{kg} \] Thus, the answer is \( 0.2 \, \text{kg} \).