Scientific Computations

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Scientific Computations · Grade 8 · Worksheet 3

1. Hana's computer processes data at a rate of 2.4 × 10⁶ bytes per second. She needs to transfer a file of size 9.6 × 10¹⁰ bytes. How many seconds will the transfer take? Answer: 40000 Solution: We know rate = 2.4 × 10⁶ bytes/second and total = 9.6 × 10¹⁰ bytes. Time = total ÷ rate = (9.6 × 10¹⁰) ÷ (2.4 × 10⁶). Divide the coefficients: 9.6 ÷ 2.4 = 4.
   Full step-by-step solution

   Step 1: We know rate = 2.4 × 10⁶ bytes/second and total = 9.6 × 10¹⁰ bytes. Step 2: Time = total ÷ rate = (9.6 × 10¹⁰) ÷ (2.4 × 10⁶). Step 3: Divide the coefficients: 9.6 ÷ 2.4 = 4. Step 4: Divide the powers of 10: 10¹⁰ ÷ 10⁶ = 10^(10-6) = 10⁴. Step 5: So time = 4 × 10⁴ = 40,000 seconds. The answer is 40000.

2. A star is 8.1 × 10¹⁶ meters from Earth. Light travels at 3.0 × 10⁸ m/s. How many seconds does it take for light from the star to reach Earth? Answer: 270000000 Solution: Write the formula: time = distance ÷ speed Substitute the values: time = (8.1 × 10¹⁶) ÷ (3.0 × 10⁸) Divide the coefficients: 8.1 ÷ 3.0 = 2.7 Subtract the exponents: 10¹⁶ ÷ 10⁸ = 10^(16-8) = 10⁸ Combine: 2.7 × 10⁸ Convert to standard form: 2.7 × 10⁸ = 270,000,000 The answer is 270000000.
   Full step-by-step solution

   Step 1: Write the formula: time = distance ÷ speed Step 2: Substitute the values: time = (8.1 × 10¹⁶) ÷ (3.0 × 10⁸) Step 3: Divide the coefficients: 8.1 ÷ 3.0 = 2.7 Step 4: Subtract the exponents: 10¹⁶ ÷ 10⁸ = 10^(16-8) = 10⁸ Step 5: Combine: 2.7 × 10⁸ Step 6: Convert to standard form: 2.7 × 10⁸ = 270,000,000 The answer is 270000000.

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. Mere's factory produces 9.6 × 10⁶ microchips per day. Each microchip has a mass of 2.5 × 10⁻⁴ grams. What is the total mass of microchips produced in one day? Answer: 2400 Solution: Multiply the coefficients: 9.6 × 2.5 = 24.0 Multiply the powers of 10: 10⁶ × 10⁻⁴ = 10^(6 + (-4)) = 10² Combine: 24.0 × 10² = 24.0 × 100 = 2400 Convert to standard form: 2400 grams The answer is 2400.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 9.6 × 2.5 = 24.0 Step 2: Multiply the powers of 10: 10⁶ × 10⁻⁴ = 10^(6 + (-4)) = 10² Step 3: Combine: 24.0 × 10² = 24.0 × 100 = 2400 Step 4: Convert to standard form: 2400 grams The answer is 2400.

5. A research satellite is traveling from Earth to Mars. The distance between the planets is approximately 2.25 × 10^8 kilometers. If the satellite travels at a constant speed of 7.5 × 10^4 kilometers per hour, how many hours will the journey take? Express your answer in scientific notation. Answer: 3.0 × 10^3 Solution: Distance = 2.25 × 10^8 km Speed = 7.5 × 10^4 km/h We need to find time in hours.
   Full step-by-step solution

   Step 1: Understand the problem We know: Distance = 2.25 × 10^8 km Speed = 7.5 × 10^4 km/h We need to find time in hours. Step 2: Recall the relationship between distance, speed, and time Time = Distance / Speed Step 3: Substitute the given values Time = (2.25 × 10^8) / (7.5 × 10^4) Step 4: Separate the decimal part and the powers of 10 Time = (2.25 / 7.5) × (10^8 / 10^4) Step 5: Simplify the decimal part 2.25 / 7.5 = 225 / 750 Divide numerator and denominator by 75: 225 ÷ 75 = 3 750 ÷ 75 = 10 So 2.25 / 7.5 = 3/10 = 0.3 Step 6: Simplify the powers of 10 10^8 / 10^4 = 10^(8 - 4) = 10^4 Step 7: Combine the results Time = 0.3 × 10^4 Step 8: Convert to proper scientific notation 0.3 × 10^4 = 3.0 × 10^(-1) × 10^4 = 3.0 × 10^(4 - 1) = 3.0 × 10^3 Step 9: Final answer The journey takes 3.0 × 10^3 hours.

6. A research satellite is traveling at a speed of 2.9 × 10⁴ kilometers per hour. If it needs to cover a distance of 1.305 × 10⁶ kilometers to reach its observation position, how many hours will the journey take? Answer: 45 Solution: Speed = 2.9 × 10⁴ km/h Distance = 1.305 × 10⁶ km Time = Distance / Speed Time = (1.305 × 10⁶) / (2.9 × 10⁴) Time = (1.305 / 2.9) × (10⁶ / 10⁴) 10⁶ / 10⁴ = 10^(6 - 4) = 10² = 100 Time = (1.305 / 2.9) × 100 Divide 1.305 by 2.9: 1.305 ÷ 2.9 Multiply numerator and denominator by 10 to make it…
   Full step-by-step solution

   We are given: Speed = 2.9 × 10⁴ km/h Distance = 1.305 × 10⁶ km Time = Distance / Speed Step 1: Write the division: Time = (1.305 × 10⁶) / (2.9 × 10⁴) Step 2: Separate the decimal part and the powers of ten: Time = (1.305 / 2.9) × (10⁶ / 10⁴) Step 3: Simplify the powers of ten: 10⁶ / 10⁴ = 10^(6 - 4) = 10² = 100 So: Time = (1.305 / 2.9) × 100 Step 4: Divide 1.305 by 2.9: 1.305 ÷ 2.9 Multiply numerator and denominator by 10 to make it easier: 13.05 ÷ 29 29 × 0.45 = 13.05 exactly. So 1.305 / 2.9 = 0.45 Step 5: Multiply by 100: 0.45 × 100 = 45 Step 6: Conclusion: The journey will take 45 hours. Final answer: 45

7. A spacecraft travels at a constant speed of 2.5 × 10⁴ kilometers per hour. How many hours would it take to travel a distance of 3.75 × 10⁸ kilometers? Write your answer in standard form. Answer: 15000 Solution: Speed = 2.5 × 10⁴ km/h Distance = 3.75 × 10⁸ km Time = Distance / Speed Time = (3.75 × 10⁸) / (2.5 × 10⁴) Separate the decimal part and the power of ten part: = (3.75 / 2.5) × (10⁸ / 10⁴) 3.75 / 2.5 = 1.5 10⁸ / 10⁴ = 10^(8 - 4) = 10⁴ 1.5 × 10⁴ 1.5 × 10⁴ = 1.5 × 10000 = 15000 Final answer: 15000…
   Full step-by-step solution

   We are given: Speed = 2.5 × 10⁴ km/h Distance = 3.75 × 10⁸ km We know: Time = Distance / Speed Step 1: Write the division: Time = (3.75 × 10⁸) / (2.5 × 10⁴) Step 2: Separate the decimal part and the power of ten part: = (3.75 / 2.5) × (10⁸ / 10⁴) Step 3: Divide the decimals: 3.75 / 2.5 = 1.5 Step 4: Divide the powers of ten: 10⁸ / 10⁴ = 10^(8 - 4) = 10⁴ Step 5: Combine results: 1.5 × 10⁴ Step 6: Convert to standard form: 1.5 × 10⁴ = 1.5 × 10000 = 15000 Final answer: 15000 hours

8. A star in the Andromeda galaxy emits (8.4 × 10²⁶) watts of power. If a space telescope collects energy for (7.2 × 10³) seconds, how many joules of energy does it collect? (Energy = Power × Time) Answer: 6.048 × 10³⁰ Solution: Write the formula: Energy = Power × Time Substitute the values: Energy = (8.4 × 10²⁶) × (7.2 × 10³) Multiply the coefficients: 8.4 × 7.2 = 60.48 Add the exponents: 10²⁶ × 10³ = 10^(26+3) = 10²⁹ Combine: 60.48 × 10²⁹ Convert to proper scientific notation: 60.48 × 10²⁹ = 6.048 × 10¹ × 10²⁹ = 6.048…
   Full step-by-step solution

   Step 1: Write the formula: Energy = Power × Time Step 2: Substitute the values: Energy = (8.4 × 10²⁶) × (7.2 × 10³) Step 3: Multiply the coefficients: 8.4 × 7.2 = 60.48 Step 4: Add the exponents: 10²⁶ × 10³ = 10^(26+3) = 10²⁹ Step 5: Combine: 60.48 × 10²⁹ Step 6: Convert to proper scientific notation: 60.48 × 10²⁹ = 6.048 × 10¹ × 10²⁹ = 6.048 × 10³⁰ The answer is 6.048 × 10³⁰ joules.

9. A rectangular solar panel measures 2.4 × 10³ millimeters in length and 8 × 10² millimeters in width. What is the area of the solar panel in square meters? (Remember: 1 meter = 1000 millimeters) Answer: 1.92 Solution: Length \( L = 2.4 \times 10^3 \) mm Width \( W = 8 \times 10^2 \) mm Area in mm² = \( L \times W \) = \( (2.4 \times 10^3) \times (8 \times 10^2) \) = \( 2.4 \times 8 \times 10^{3+2} \) = \( 19.2 \times 10^5 \) mm² We know: 1 meter = 1000 millimeters So 1 m² = (1000 mm) × (1000 mm) = \( 10^6 \)…
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Write down the given dimensions in millimeters** Length \( L = 2.4 \times 10^3 \) mm Width \( W = 8 \times 10^2 \) mm --- **Step 2: Calculate the area in square millimeters** Area in mm² = \( L \times W \) = \( (2.4 \times 10^3) \times (8 \times 10^2) \) = \( 2.4 \times 8 \times 10^{3+2} \) = \( 19.2 \times 10^5 \) mm² --- **Step 3: Convert square millimeters to square meters** We know: 1 meter = 1000 millimeters So 1 m² = (1000 mm) × (1000 mm) = \( 10^6 \) mm² Therefore: Area in m² = \( \frac{19.2 \times 10^5}{10^6} \) = \( 19.2 \times 10^{5-6} \) = \( 19.2 \times 10^{-1} \) = \( 1.92 \) --- **Step 4: Final answer** The area of the solar panel is \( 1.92 \) square meters.