Scientific Computations

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Scientific Computations · Grade 8 · Worksheet 1

1. A research satellite is traveling at a speed of 2.8 × 10⁴ kilometers per hour. If it needs to cover a distance of 1.68 × 10⁶ kilometers to reach its observation position, how many hours will the journey take? Answer: 60 Solution: Speed = 2.8 × 10⁴ km/h Distance = 1.68 × 10⁶ km Time = ? Recall the formula for time. Time = Distance / Speed Substitute the given values.
   Full step-by-step solution

   We are given: Speed = 2.8 × 10⁴ km/h Distance = 1.68 × 10⁶ km Time = ? Step 1: Recall the formula for time. Time = Distance / Speed Step 2: Substitute the given values. Time = (1.68 × 10⁶) / (2.8 × 10⁴) Step 3: Separate the decimal part and the power of ten part. Time = (1.68 / 2.8) × (10⁶ / 10⁴) Step 4: Simplify the decimal division. 1.68 / 2.8 = 168 / 280 Divide numerator and denominator by 56: 168 ÷ 56 = 3 280 ÷ 56 = 5 So 1.68 / 2.8 = 3/5 = 0.6 Step 5: Simplify the powers of ten. 10⁶ / 10⁴ = 10^(6 - 4) = 10² = 100 Step 6: Multiply the results. Time = 0.6 × 100 = 60 Step 7: State the final answer. The journey will take 60 hours.

2. 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

3. Mere's warehouse ships 4.8 × 10⁶ kilograms of rice per month. If each shipment weighs 6.0 × 10² kilograms, how many shipments are made each month? Answer: 8000 Solution: We need to divide the total rice by the weight per shipment: (4.8 × 10⁶) ÷ (6.0 × 10²) Divide the coefficients: 4.8 ÷ 6.0 = 0.8 Divide the powers of ten: 10⁶ ÷ 10² = 10⁶⁻² = 10⁴ Combine: 0.8 × 10⁴ Convert to standard form: 0.8 × 10⁴ = 8 × 10³ = 8000 The answer is 8000.
   Full step-by-step solution

   Step 1: We need to divide the total rice by the weight per shipment: (4.8 × 10⁶) ÷ (6.0 × 10²) Step 2: Divide the coefficients: 4.8 ÷ 6.0 = 0.8 Step 3: Divide the powers of ten: 10⁶ ÷ 10² = 10⁶⁻² = 10⁴ Step 4: Combine: 0.8 × 10⁴ Step 5: Convert to standard form: 0.8 × 10⁴ = 8 × 10³ = 8000 The answer is 8000.

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 lab is studying bacteria growth. They observe that a single bacterium divides into 2 every 30 minutes. If the lab starts with 5.0 × 10³ bacteria in a petri dish, how many bacteria will there be after 4 hours? Express your answer in scientific notation. Answer: 1.28 × 10⁶ Solution: We start with \( N_0 = 5.0 \times 10^3 \) bacteria. Each bacterium divides into 2 every 30 minutes, so the doubling time is 30 minutes. We want the number after 4 hours.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We start with \( N_0 = 5.0 \times 10^3 \) bacteria. Each bacterium divides into 2 every 30 minutes, so the doubling time is 30 minutes. We want the number after 4 hours. --- **Step 2: Find the number of doubling periods in 4 hours** Doubling time = 30 minutes = 0.5 hours. Total time = 4 hours. Number of doubling periods \( n \) = total time / doubling time \( n = 4 / 0.5 = 8 \) So there are 8 doublings. --- **Step 3: Use the exponential growth formula** For doubling: \( N = N_0 \times 2^n \) Substitute: \( N = (5.0 \times 10^3) \times 2^8 \) --- **Step 4: Calculate \( 2^8 \)** \( 2^8 = 256 \) So: \( N = 5.0 \times 10^3 \times 256 \) --- **Step 5: Multiply** First multiply 5.0 × 256: \( 5.0 \times 256 = 1280 \) So: \( N = 1280 \times 10^3 \) --- **Step 6: Convert to scientific notation** \( 1280 \times 10^3 = 1.28 \times 10^3 \times 10^3 \) \( = 1.28 \times 10^{3+3} \) \( = 1.28 \times 10^6 \) --- **Final Answer:** \( 1.28 \times 10^6 \) bacteria

6. A rectangular solar panel has dimensions of 3.6 × 10² cm by 2.5 × 10³ cm. If you arrange these panels in a grid with 4 rows and 6 columns to create a solar array, what is the total area of the array in scientific notation? Answer: 2.16 × 10⁷ cm² Solution: Length = 3.6 × 10² cm Width = 2.5 × 10³ cm Area of one panel = Length × Width = (3.6 × 10²) × (2.5 × 10³) Multiply the decimal parts: 3.6 × 2.5 = 9.0 Multiply the powers of ten: 10² × 10³ = 10^(2+3) = 10⁵ So area of one panel = 9.0 × 10⁵ cm² The array has 4 rows and 6 columns.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the area of one solar panel** The panel dimensions are: Length = 3.6 × 10² cm Width = 2.5 × 10³ cm Area of one panel = Length × Width = (3.6 × 10²) × (2.5 × 10³) Multiply the decimal parts: 3.6 × 2.5 = 9.0 Multiply the powers of ten: 10² × 10³ = 10^(2+3) = 10⁵ So area of one panel = 9.0 × 10⁵ cm² --- **Step 2: Determine the number of panels in the array** The array has 4 rows and 6 columns. Total panels = 4 × 6 = 24 panels --- **Step 3: Find the total area of the array** Total area = (Area of one panel) × (Number of panels) = (9.0 × 10⁵) × 24 First, multiply 9.0 × 24 = 216 So 216 × 10⁵ cm² --- **Step 4: Convert to scientific notation** 216 = 2.16 × 10² So 216 × 10⁵ = (2.16 × 10²) × 10⁵ = 2.16 × 10^(2+5) = 2.16 × 10⁷ cm² --- **Final Answer:** 2.16 × 10⁷ cm²

7. A star in the Andromeda galaxy emits 1.6 × 10²⁶ watts of energy. If a nearby planet absorbs 6.4 × 10⁻⁶ of this energy, how many watts does the planet absorb? Answer: 1.024 × 10²¹ Solution: The planet absorbs 6.4 × 10⁻⁶ of the star's energy. This means we multiply: (1.6 × 10²⁶) × (6.4 × 10⁻⁶). Multiply the coefficients: 1.6 × 6.4 = 10.24.
   Full step-by-step solution

   Step 1: The planet absorbs 6.4 × 10⁻⁶ of the star's energy. This means we multiply: (1.6 × 10²⁶) × (6.4 × 10⁻⁶). Step 2: Multiply the coefficients: 1.6 × 6.4 = 10.24. Step 3: Multiply the powers of 10: 10²⁶ × 10⁻⁶ = 10²⁶⁺⁽⁻⁶⁾ = 10²⁰. Step 4: Combine: 10.24 × 10²⁰. Step 5: Convert to proper scientific notation: 10.24 = 1.024 × 10¹, so 1.024 × 10¹ × 10²⁰ = 1.024 × 10²¹. The answer is 1.024 × 10²¹ watts.

8. A research lab is studying nanotechnology. They create nanoparticles that are 2.5 × 10⁻⁸ meters in diameter. If they arrange these nanoparticles in a straight line to create a nanoscale circuit that is 1.5 × 10⁻⁴ meters long, how many nanoparticles are needed? Express your answer in scientific notation. Answer: 6.0 × 10³ Solution: Identify the total length needed: 1.5 × 10⁻⁴ meters Identify the length of one nanoparticle: 2.5 × 10⁻⁸ meters Calculate the number of nanoparticles by dividing total length by length per nanoparticle: (1.5 × 10⁻⁴) ÷ (2.5 × 10⁻⁸) Divide the coefficients: 1.5 ÷ 2.5 = 0.6 Divide the powers of 10:…
   Full step-by-step solution

   Step 1: Identify the total length needed: 1.5 × 10⁻⁴ meters Step 2: Identify the length of one nanoparticle: 2.5 × 10⁻⁸ meters Step 3: Calculate the number of nanoparticles by dividing total length by length per nanoparticle: (1.5 × 10⁻⁴) ÷ (2.5 × 10⁻⁸) Step 4: Divide the coefficients: 1.5 ÷ 2.5 = 0.6 Step 5: Divide the powers of 10: 10⁻⁴ ÷ 10⁻⁸ = 10⁻⁴⁻⁽⁻⁸⁾ = 10⁴ Step 6: Multiply the results: 0.6 × 10⁴ = 6.0 × 10³ The answer is 6.0 × 10³ nanoparticles.