Scientific Computations

lessonbunny.com

Answer Key & Explanations

Scientific Computations · Grade 8 · Worksheet 1

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. Olivia's solar farm generates 7.5 × 10⁶ watts of power. If each household requires 1.5 × 10³ watts, how many households can the farm power? Answer: 5000 Solution: We need to divide the total power by the power per household: (7.5 × 10⁶) ÷ (1.5 × 10³). Divide the coefficients: 7.5 ÷ 1.5 = 5. Divide the powers of ten: 10⁶ ÷ 10³ = 10^(6-3) = 10³.
   Full step-by-step solution

   Step 1: We need to divide the total power by the power per household: (7.5 × 10⁶) ÷ (1.5 × 10³). Step 2: Divide the coefficients: 7.5 ÷ 1.5 = 5. Step 3: Divide the powers of ten: 10⁶ ÷ 10³ = 10^(6-3) = 10³. Step 4: Multiply the results: 5 × 10³ = 5000. The answer is 5000.

3. Mason is designing a rectangular garden in the shape of a grid for a science project. The garden is divided into 8 rows and 12 columns of identical square plots. Each plot has a side length of 2.5 × 10² millimeters. What is the total area of the entire garden in square millimeters? Express your answer in scientific notation. Answer: 6.0 × 10⁷ Solution: Find the area of one square plot. Area of one plot = side × side = (2.5 × 10²) × (2.5 × 10²) = (2.5 × 2.5) × (10² × 10²) = 6.25 × 10⁴ square millimeters. Find the total number of plots.
   Full step-by-step solution

   Step 1: Find the area of one square plot. Area of one plot = side × side = (2.5 × 10²) × (2.5 × 10²) = (2.5 × 2.5) × (10² × 10²) = 6.25 × 10⁴ square millimeters. Step 2: Find the total number of plots. Total plots = rows × columns = 8 × 12 = 96 plots. Step 3: Find the total area of the garden. Total area = area of one plot × total number of plots = (6.25 × 10⁴) × 96. First, multiply the coefficients: 6.25 × 96 = 600. So, total area = 600 × 10⁴ square millimeters. Step 4: Convert to proper scientific notation. 600 × 10⁴ = 6.0 × 10² × 10⁴ = 6.0 × 10⁶ square millimeters. The answer is 6.0 × 10⁷ square millimeters.

4. Olivia is an astrophysicist studying a supernova remnant. She calculates that the remnant has a mass of 5.0 × 10^30 kilograms and is expanding outward at a speed of 8.0 × 10^6 meters per second. What is the momentum of the supernova remnant in kilogram-meters per second? (Momentum = mass × velocity) Express your answer in scientific notation. Answer: 4.0 × 10^37 Solution: Write the formula for momentum: p = m × v Identify the values: mass = 5.0 × 10^30 kg, velocity = 8.0 × 10^6 m/s Multiply the coefficients: 5.0 × 8.0 = 40.0 Multiply the powers of 10: 10^30 × 10^6 = 10^(30+6) = 10^36 Combine: 40.0 × 10^36 Adjust to proper scientific notation (coefficient between…
   Full step-by-step solution

   Step 1: Write the formula for momentum: p = m × v Step 2: Identify the values: mass = 5.0 × 10^30 kg, velocity = 8.0 × 10^6 m/s Step 3: Multiply the coefficients: 5.0 × 8.0 = 40.0 Step 4: Multiply the powers of 10: 10^30 × 10^6 = 10^(30+6) = 10^36 Step 5: Combine: 40.0 × 10^36 Step 6: Adjust to proper scientific notation (coefficient between 1 and 10): 40.0 × 10^36 = 4.0 × 10^37 The answer is 4.0 × 10^37 kg·m/s.

5. 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.

6. Kaia is studying a microscopic crystal structure under a microscope. The crystal is composed of identical rectangular prism-shaped unit cells arranged in a grid. The grid has 3.5 × 10³ unit cells along the length and 7.1 × 10² unit cells along the width. Each unit cell has dimensions of 5.0 × 10⁻⁹ meters by 3.0 × 10⁻⁹ meters. If the unit cells are placed edge-to-edge with no gaps, what is the total area covered by the crystal grid in square meters? Express your answer in scientific notation. Answer: 3.7275 × 10⁻² Solution: Find the total number of unit cells. Total unit cells = (3.5 × 10³) × (7.1 × 10²) = 3.5 × 7.1 × 10³⁺² = 24.85 × 10⁵ = 2.485 × 10⁶ unit cells (adjusting to proper scientific notation) Find the area of one unit cell.
   Full step-by-step solution

   Step 1: Find the total number of unit cells. Total unit cells = (3.5 × 10³) × (7.1 × 10²) = 3.5 × 7.1 × 10³⁺² = 24.85 × 10⁵ = 2.485 × 10⁶ unit cells (adjusting to proper scientific notation) Step 2: Find the area of one unit cell. Area of one unit cell = (5.0 × 10⁻⁹ m) × (3.0 × 10⁻⁹ m) = 5.0 × 3.0 × 10⁻⁹⁻⁹ = 15 × 10⁻¹⁸ = 1.5 × 10⁻¹⁷ m² (adjusting to proper scientific notation) Step 3: Find the total area of the crystal grid. Total area = (2.485 × 10⁶) × (1.5 × 10⁻¹⁷) = 2.485 × 1.5 × 10⁶⁻¹⁷ = 3.7275 × 10⁻¹¹ m² The answer is 3.7275 × 10⁻¹¹ square meters.

7. A research satellite is studying the Andromeda Galaxy, which is approximately 2.4 × 10^19 kilometers from Earth. The satellite transmits high-resolution images that require 8.0 × 10^12 bytes of data each. If the transmission speed is 1.6 × 10^7 bytes per second, how many seconds does it take to transmit one complete image? Express your answer in scientific notation. Answer: 5.0 × 10^5 Solution: Identify the total data to transmit: 8.0 × 10^12 bytes Identify the transmission speed: 1.6 × 10^7 bytes per second Calculate time using the formula: time = total data / transmission speed time = (8.0 × 10^12) / (1.6 × 10^7) Divide the coefficients: 8.0 / 1.6 = 5.0 Subtract the exponents: 12 - 7…
   Full step-by-step solution

   Step 1: Identify the total data to transmit: 8.0 × 10^12 bytes Step 2: Identify the transmission speed: 1.6 × 10^7 bytes per second Step 3: Calculate time using the formula: time = total data / transmission speed Step 4: time = (8.0 × 10^12) / (1.6 × 10^7) Step 5: Divide the coefficients: 8.0 / 1.6 = 5.0 Step 6: Subtract the exponents: 12 - 7 = 5 Step 7: Combine the results: 5.0 × 10^5 seconds The answer is 5.0 × 10^5 seconds.