Scientific Computations

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Scientific Computations · Grade 8 · Worksheet 2

1. A rectangular solar panel array is arranged in a grid with 4 rows and 5 columns. Each panel measures 1.8 × 10² cm by 9 × 10¹ cm. If the panels are placed edge-to-edge with no gaps, what is the total area covered by the array in square centimeters? Express your answer in scientific notation. Answer: 3.24 × 10⁵ Solution: Area = length × width = (1.8 × 10² cm) × (9 × 10¹ cm) = (1.8 × 9) × (10² × 10¹) = 16.2 × 10³ = 1.62 × 10⁴ cm² Total panels = rows × columns = 4 × 5 = 20 panels Total area = area of one panel × number of panels = (1.62 × 10⁴ cm²) × 20 = (1.62 × 10⁴) × (2 × 10¹) = (1.62 × 2) × (10⁴ × 10¹) = 3.24 ×…
   Full step-by-step solution

   Step 1: Find the area of one panel Area = length × width = (1.8 × 10² cm) × (9 × 10¹ cm) = (1.8 × 9) × (10² × 10¹) = 16.2 × 10³ = 1.62 × 10⁴ cm² Step 2: Find the total number of panels Total panels = rows × columns = 4 × 5 = 20 panels Step 3: Find the total area of the array Total area = area of one panel × number of panels = (1.62 × 10⁴ cm²) × 20 = (1.62 × 10⁴) × (2 × 10¹) = (1.62 × 2) × (10⁴ × 10¹) = 3.24 × 10⁵ cm² The answer is 3.24 × 10⁵.

2. A rectangular solar panel array is arranged in a grid pattern with 3.2 × 10² panels along the length and 1.5 × 10³ panels along the width. Each panel measures 1.6 meters by 0.8 meters. If the panels are placed edge-to-edge with no gaps, what is the total area covered by the array in square meters? Express your answer in scientific notation. Answer: 6.144 × 10⁵ Solution: - Number of panels along the length: \( 3.2 \times 10^2 = 320 \) panels - Number of panels along the width: \( 1.5 \times 10^3 = 1500 \) panels - Each panel is 1.6 m by 0.8 m.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular grid of solar panels: - Number of panels along the length: \( 3.2 \times 10^2 = 320 \) panels - Number of panels along the width: \( 1.5 \times 10^3 = 1500 \) panels - Each panel is 1.6 m by 0.8 m. - Panels are placed edge-to-edge with no gaps. We want the **total area covered** in square meters, in scientific notation. --- **Step 2: Find the area of one panel** Area of one panel = length × width = \( 1.6 \, \text{m} \times 0.8 \, \text{m} \) = \( 1.28 \, \text{m}^2 \) --- **Step 3: Find the total number of panels** Total panels = (panels along length) × (panels along width) = \( 320 \times 1500 \) First: \( 320 \times 1500 \) = \( 320 \times (1.5 \times 10^3) \) = \( 480 \times 10^3 \) = \( 4.8 \times 10^5 \) panels --- **Step 4: Find total area** Total area = (number of panels) × (area per panel) = \( (4.8 \times 10^5) \times 1.28 \) Multiply \( 4.8 \times 1.28 \): \( 4.8 \times 1 = 4.8 \) \( 4.8 \times 0.28 = 4.8 \times (0.2 + 0.08) = 0.96 + 0.384 = 1.344 \) So \( 4.8 + 1.344 = 6.144 \) Thus: \( 4.8 \times 1.28 = 6.144 \) Now multiply by \( 10^5 \): \( 6.144 \times 10^5 \, \text{m}^2 \) --- **Step 5: Verify units and scientific notation** Area is in square meters, and \( 6.144 \times 10^5 \) is in scientific notation (one digit before the decimal). --- **Final Answer:** \( 6.144 \times 10^5 \)

3. Emma's factory produces 3.5 × 10⁵ light bulbs per day. Each bulb requires 4.0 × 10⁻³ kg of glass. How many kilograms of glass are needed for 5 days of production? Answer: 7000 Solution: Find total bulbs in 5 days: (3.5 × 10⁵) × 5 = 17.5 × 10⁵ = 1.75 × 10⁶ bulbs. Multiply total bulbs by glass per bulb: (1.75 × 10⁶) × (4.0 × 10⁻³). Multiply coefficients: 1.75 × 4.0 = 7.0.
   Full step-by-step solution

   Step 1: Find total bulbs in 5 days: (3.5 × 10⁵) × 5 = 17.5 × 10⁵ = 1.75 × 10⁶ bulbs. Step 2: Multiply total bulbs by glass per bulb: (1.75 × 10⁶) × (4.0 × 10⁻³). Step 3: Multiply coefficients: 1.75 × 4.0 = 7.0. Step 4: Add exponents: 10⁶ × 10⁻³ = 10³. Step 5: Result: 7.0 × 10³ = 7000. The answer is 7000.

4. Olivia is an environmental scientist studying the water cycle in a rainforest. She calculates that on average, a single large tree releases 9.5 × 10^3 liters of water vapor into the atmosphere per day. In a section of the rainforest containing 7 × 10^5 similar trees, what is the total volume of water vapor released by all these trees in one day? Express your answer in scientific notation. Answer: 6.65 × 10^9 liters Solution: Identify the amount of water vapor per tree: 9.5 × 10^3 liters/day Identify the number of trees: 7 × 10^5 trees Multiply to find the total volume: (9.5 × 10^3) × (7 × 10^5) Multiply the coefficients: 9.5 × 7 = 66.5 Multiply the powers of 10: 10^3 × 10^5 = 10^8 Combine: 66.5 × 10^8 Convert to…
   Full step-by-step solution

   Step 1: Identify the amount of water vapor per tree: 9.5 × 10^3 liters/day Step 2: Identify the number of trees: 7 × 10^5 trees Step 3: Multiply to find the total volume: (9.5 × 10^3) × (7 × 10^5) Step 4: Multiply the coefficients: 9.5 × 7 = 66.5 Step 5: Multiply the powers of 10: 10^3 × 10^5 = 10^8 Step 6: Combine: 66.5 × 10^8 Step 7: Convert to proper scientific notation (coefficient between 1 and 10): 6.65 × 10^9 The answer is 6.65 × 10^9 liters.

5. 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

6. A scientist is studying bacteria growth. The initial population is 2.5 × 10^6 bacteria. After 3 hours, the population has increased by a factor of 8 × 10^2. What is the new population of bacteria? Express your answer in scientific notation. Answer: 2e9 Solution: Initial population = 2.5 × 10^6 The problem says the population increased by a factor of 8 × 10^2. New population = Initial population × (8 × 10^2) First, multiply the decimal parts and the powers of 10 separately.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Write down the initial population** Initial population = 2.5 × 10^6 --- **Step 2: Understand the increase factor** The problem says the population increased by a factor of 8 × 10^2. That means: New population = Initial population × (8 × 10^2) --- **Step 3: Multiply the numbers** First, multiply the decimal parts and the powers of 10 separately. Initial population = 2.5 × 10^6 Factor = 8 × 10^2 Multiply 2.5 × 8 = 20 Multiply the powers of 10: 10^6 × 10^2 = 10^(6+2) = 10^8 So we have: 20 × 10^8 --- **Step 4: Convert to proper scientific notation** 20 × 10^8 is not in standard scientific notation because 20 is not between 1 and 10. 20 = 2.0 × 10^1 So: 20 × 10^8 = (2.0 × 10^1) × 10^8 = 2.0 × 10^(1+8) = 2.0 × 10^9 --- **Step 5: Write the final answer** 2.0 × 10^9 can be written as 2e9. --- **Final answer:** 2e9