Scientific Computations

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Scientific Computations · Grade 8 · Worksheet 2

1. A rectangular solar farm is arranged in a grid pattern with 2.5 × 10² panels along the length and 4.8 × 10² panels along the width. Each individual panel measures 1.2 meters by 2.0 meters. If the panels are placed edge-to-edge with no gaps, what is the total area covered by the solar farm in square meters? Express your answer in scientific notation. Answer: 2.88 × 10⁵ Solution: Area of one panel = length × width = 1.2 m × 2.0 m = 2.4 m² Total panels = panels along length × panels along width Total panels = (2.5 × 10²) × (4.8 × 10²) = (2.5 × 4.8) × (10² × 10²) = 12 × 10⁴ Convert 12 × 10⁴ to proper scientific notation 12 × 10⁴ = 1.2 × 10 × 10⁴ = 1.2 × 10⁵ panels Total…
   Full step-by-step solution

   Step 1: Calculate the area of one solar panel Area of one panel = length × width = 1.2 m × 2.0 m = 2.4 m² Step 2: Calculate the total number of panels in the grid Total panels = panels along length × panels along width Total panels = (2.5 × 10²) × (4.8 × 10²) = (2.5 × 4.8) × (10² × 10²) = 12 × 10⁴ Step 3: Convert 12 × 10⁴ to proper scientific notation 12 × 10⁴ = 1.2 × 10 × 10⁴ = 1.2 × 10⁵ panels Step 4: Calculate the total area of the solar farm Total area = area of one panel × total number of panels Total area = 2.4 m² × 1.2 × 10⁵ = (2.4 × 1.2) × 10⁵ = 2.88 × 10⁵ m² The answer is 2.88 × 10⁵.

2. A research satellite is studying distant galaxies. It detects a star cluster approximately 4.2 × 10^18 kilometers away. The satellite's camera has a resolution that can distinguish objects separated by 7.5 × 10^14 kilometers. What is the maximum number of distinct resolution units between Earth and this star cluster? Express your answer in scientific notation. Answer: 5.6 × 10^3 Solution: We need the maximum number of distinct resolution units between Earth and the star cluster. This means we divide the total distance by the smallest distance the camera can resolve. Total distance to star cluster = 4.2 × 10^18 km Resolution (smallest distinguishable separation) = 7.5 × 10^14 km…
   Full step-by-step solution

   Step 1: Understand the problem We need the maximum number of distinct resolution units between Earth and the star cluster. This means we divide the total distance by the smallest distance the camera can resolve. Step 2: Write the given numbers Total distance to star cluster = 4.2 × 10^18 km Resolution (smallest distinguishable separation) = 7.5 × 10^14 km Step 3: Set up the division Number of resolution units = (Total distance) / (Resolution distance) = (4.2 × 10^18) / (7.5 × 10^14) Step 4: Separate the coefficients and powers of 10 = (4.2 / 7.5) × (10^18 / 10^14) Step 5: Simplify the powers of 10 10^18 / 10^14 = 10^(18 - 14) = 10^4 Step 6: Simplify the coefficient 4.2 / 7.5 = 42 / 75 (multiplying numerator and denominator by 10) 42 / 75 = (42 ÷ 3) / (75 ÷ 3) = 14 / 25 14 / 25 = 0.56 Step 7: Combine results 0.56 × 10^4 Step 8: Write in correct scientific notation 0.56 × 10^4 = 5.6 × 10^3 Step 9: Interpret the answer The maximum number of distinct resolution units between Earth and the star cluster is 5.6 × 10^3. Final answer: 5.6 × 10^3

3. Noah's factory produces 7.2 × 10⁹ microchips per year. Each microchip has a mass of 8.5 × 10⁻⁶ grams. What is the total mass of microchips produced in one year? Answer: 61200 Solution: Multiply the coefficients: 7.2 × 8.5 = 61.2 Add the exponents: 10⁹ × 10⁻⁶ = 10^(9 + (-6)) = 10³ Combine: 61.2 × 10³ Convert to proper scientific notation: 61.2 × 10³ = 6.12 × 10⁴ Convert to standard form: 6.12 × 10⁴ = 61,200 The answer is 61200.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 7.2 × 8.5 = 61.2 Step 2: Add the exponents: 10⁹ × 10⁻⁶ = 10^(9 + (-6)) = 10³ Step 3: Combine: 61.2 × 10³ Step 4: Convert to proper scientific notation: 61.2 × 10³ = 6.12 × 10⁴ Step 5: Convert to standard form: 6.12 × 10⁴ = 61,200 The answer is 61200.

4. A star is 7.5 × 10⁹ km from Earth. Light travels at 3.0 × 10⁵ km/s. How many seconds does it take for light from the star to reach Earth? Answer: 25000 Solution: Write the formula: time = distance ÷ speed. Substitute the values: time = (7.5 × 10⁹) ÷ (3.0 × 10⁵). Divide the coefficients: 7.5 ÷ 3.0 = 2.5.
   Full step-by-step solution

   Step 1: Write the formula: time = distance ÷ speed. Step 2: Substitute the values: time = (7.5 × 10⁹) ÷ (3.0 × 10⁵). Step 3: Divide the coefficients: 7.5 ÷ 3.0 = 2.5. Step 4: Divide the powers of 10: 10⁹ ÷ 10⁵ = 10^(9-5) = 10⁴. Step 5: Combine: 2.5 × 10⁴ = 25,000 seconds. The answer is 25000.

5. A research satellite is traveling at a speed of 2.9 × 10⁴ kilometers per hour. If it needs to cover a distance of 1.45 × 10⁶ kilometers to reach its observation position, how many hours will the journey take? Answer: 50 Solution: Speed = 2.9 × 10⁴ km/h Distance = 1.45 × 10⁶ km We need to find the time in hours. Recall the formula for time. Time = Distance / Speed Substitute the given values.
   Full step-by-step solution

   We are given: Speed = 2.9 × 10⁴ km/h Distance = 1.45 × 10⁶ km We need to find the time in hours. Step 1: Recall the formula for time. Time = Distance / Speed Step 2: Substitute the given values. Time = (1.45 × 10⁶) / (2.9 × 10⁴) Step 3: Separate the decimal part and the power of ten part. Time = (1.45 / 2.9) × (10⁶ / 10⁴) Step 4: Simplify 1.45 / 2.9. 1.45 ÷ 2.9 = 145 / 290 = 29 / 58 = 1 / 2 = 0.5 Step 5: Simplify the powers of ten. 10⁶ / 10⁴ = 10^(6 - 4) = 10² = 100 Step 6: Multiply the results. 0.5 × 100 = 50 Step 7: Conclusion. The journey will take 50 hours.

6. A research satellite is traveling from Earth to Mars. The distance between the planets is approximately 5.46 × 10^7 kilometers. If the satellite travels at a constant speed of 6.5 × 10^4 kilometers per day, how many days will the journey take? Express your answer in scientific notation. Answer: 8.4 × 10^2 Solution: Distance = 5.46 × 10^7 km Speed = 6.5 × 10^4 km/day We want time in days. Recall the formula for time. Time = Distance / Speed Substitute the given values.
   Full step-by-step solution

   We are given: Distance = 5.46 × 10^7 km Speed = 6.5 × 10^4 km/day We want time in days. Step 1: Recall the formula for time. Time = Distance / Speed Step 2: Substitute the given values. Time = (5.46 × 10^7) / (6.5 × 10^4) Step 3: Separate the decimal part and the powers of 10. Time = (5.46 / 6.5) × (10^7 / 10^4) Step 4: Simplify the powers of 10. 10^7 / 10^4 = 10^(7 - 4) = 10^3 Step 5: Divide the decimals. 5.46 / 6.5 We can multiply numerator and denominator by 10 to make it easier: 54.6 / 65 Now, 65 × 0.8 = 52.0 Subtract: 54.6 - 52.0 = 2.6 65 × 0.04 = 2.6 So total = 0.8 + 0.04 = 0.84 Thus, 5.46 / 6.5 = 0.84 Step 6: Combine results. Time = 0.84 × 10^3 Step 7: Write in scientific notation. 0.84 × 10^3 = 8.4 × 10^2 Final answer: 8.4 × 10^2 days