Scientific Computations

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Scientific Computations · Grade 8 · Worksheet 3

1. A research lab is studying nanotechnology. They create nanoparticles that are each 8.4 × 10⁻⁹ meters in diameter. If they arrange these nanoparticles in a straight line to create a nanoscale circuit that is 1.68 × 10⁻⁴ meters long, how many nanoparticles are needed? Express your answer in scientific notation. Answer: 2.0 × 10⁴ Solution: Identify the total length needed: 1.68 × 10⁻⁴ meters Identify the length of one nanoparticle: 8.4 × 10⁻⁹ meters Calculate the number of nanoparticles by dividing total length by length per nanoparticle: (1.68 × 10⁻⁴) ÷ (8.4 × 10⁻⁹) Divide the coefficients: 1.68 ÷ 8.4 = 0.2 Divide the powers of…
   Full step-by-step solution

   Step 1: Identify the total length needed: 1.68 × 10⁻⁴ meters Step 2: Identify the length of one nanoparticle: 8.4 × 10⁻⁹ meters Step 3: Calculate the number of nanoparticles by dividing total length by length per nanoparticle: (1.68 × 10⁻⁴) ÷ (8.4 × 10⁻⁹) Step 4: Divide the coefficients: 1.68 ÷ 8.4 = 0.2 Step 5: Divide the powers of 10: 10⁻⁴ ÷ 10⁻⁹ = 10⁻⁴⁻⁽⁻⁹⁾ = 10⁻⁴⁺⁹ = 10⁵ Step 6: Combine results: 0.2 × 10⁵ = 2.0 × 10⁴ The answer is 2.0 × 10⁴.

2. Lena is an environmental scientist studying microplastic pollution in a large lake. She estimates there are approximately 8.4 × 10^12 microplastic particles in the lake, which has a volume of 7.5 × 10^10 cubic meters. What is the average concentration of microplastic particles per cubic meter of water? Express your answer in scientific notation. Answer: 1.12 × 10^2 Solution: Step 1: Identify the total number of particles: 8.4 × 10^12 Step 2: Identify the total volume: 7.5 × 10^10 cubic meters Step 3: Calculate concentration by dividing particles by volume: (8.4 × 10^12) ÷ (7.5 × 10^10) Step 4: Divide the coefficients: 8.4 ÷ 7.5 = 1.12 Step 5: Subtract the exponents:…
   Full step-by-step solution

   Step 1: Identify the total number of particles: 8.4 × 10^12 Step 2: Identify the total volume: 7.5 × 10^10 cubic meters Step 3: Calculate concentration by dividing particles by volume: (8.4 × 10^12) ÷ (7.5 × 10^10) Step 4: Divide the coefficients: 8.4 ÷ 7.5 = 1.12 Step 5: Subtract the exponents: 12 - 10 = 2 Step 6: Combine the results: 1.12 × 10^2 Step 7: Verify the answer represents 112 particles per cubic meter The final answer is 1.12 × 10^2.

3. Noah is an astronomer studying a newly discovered asteroid. The asteroid has a mass of 6.1 × 10^16 kilograms and is traveling at a speed of 1.6 × 10^4 meters per second. What is the kinetic energy of the asteroid in joules? (Kinetic energy = 1/2 × mass × speed^2) Express your answer in scientific notation. Answer: 7.808 × 10^24 Solution: Write the formula for kinetic energy: KE = 1/2 × m × v^2 Identify the values: mass = 6.1 × 10^16 kg, speed = 1.6 × 10^4 m/s Square the speed: (1.6 × 10^4)^2 = 1.6^2 × (10^4)^2 = 2.56 × 10^8 Multiply the mass by the squared speed: (6.1 × 10^16) × (2.56 × 10^8) = (6.1 × 2.56) × (10^16 × 10^8) =…
   Full step-by-step solution

   Step 1: Write the formula for kinetic energy: KE = 1/2 × m × v^2 Step 2: Identify the values: mass = 6.1 × 10^16 kg, speed = 1.6 × 10^4 m/s Step 3: Square the speed: (1.6 × 10^4)^2 = 1.6^2 × (10^4)^2 = 2.56 × 10^8 Step 4: Multiply the mass by the squared speed: (6.1 × 10^16) × (2.56 × 10^8) = (6.1 × 2.56) × (10^16 × 10^8) = 15.616 × 10^24 Step 5: Multiply by 1/2: (1/2) × 15.616 × 10^24 = 7.808 × 10^24 Step 6: The result is already in proper scientific notation (coefficient between 1 and 10). The answer is 7.808 × 10^24 joules.

4. A research lab is studying bacteria growth. A scientist observes that a particular bacterial culture doubles in size every 3 hours. If the initial population is 2.5 × 10³ bacteria, write an exponential function in the form y = a(b)^x that models the population after x hours. Then use this function to determine the population after 12 hours, expressing your final answer in scientific notation. Answer: 4.0 × 10⁴ Solution: Identify the initial population. The problem says the initial population is 2.5 × 10³ bacteria. So, a = 2.5 × 10³.
   Full step-by-step solution

   Step 1: Identify the initial population. The problem says the initial population is 2.5 × 10³ bacteria. So, a = 2.5 × 10³. Step 2: Determine the growth factor b. The bacteria double every 3 hours. If the doubling time is 3 hours, then b = 2 when x is measured in 3-hour periods. But in the function y = a(b)^x, x is in hours, so we must adjust b. Step 3: Write the exponential function with doubling period. General form: y = a × (2)^(x / T) where T is the doubling time in hours. Here T = 3 hours. So y = (2.5 × 10³) × (2)^(x / 3). Step 4: Match to the form y = a(b)^x. In y = a(b)^x, b is the base for hourly growth. Since doubling every 3 hours means: b^3 = 2, so b = 2^(1/3). Thus y = (2.5 × 10³) × (2^(1/3))^x But it’s simpler to keep it as y = (2.5 × 10³) × 2^(x/3). This matches y = a(b)^x if b = 2^(1/3), but for calculation we can use 2^(x/3) directly. Step 5: Find population after 12 hours. Plug x = 12 into y = (2.5 × 10³) × 2^(12/3). 12/3 = 4, so y = (2.5 × 10³) × 2^4. 2^4 = 16. So y = 2.5 × 10³ × 16. Step 6: Multiply. 2.5 × 16 = 40. So y = 40 × 10³ = 4.0 × 10⁴. Step 7: Final answer in scientific notation. The population after 12 hours is 4.0 × 10⁴ bacteria. This matches the correct answer.

5. Aroha measures the mass of a single grain of sand as 9.2 × 10⁻⁵ grams. If a beach sample contains 5.0 × 10⁶ grains of sand, what is the total mass of the sample in grams? Answer: 460 Solution: Multiply the mass of one grain by the number of grains: (9.2 × 10⁻⁵) × (5.0 × 10⁶). Multiply the coefficients: 9.2 × 5.0 = 46. Add the exponents: 10⁻⁵ × 10⁶ = 10⁻⁵⁺⁶ = 10¹.
   Full step-by-step solution

   Step 1: Multiply the mass of one grain by the number of grains: (9.2 × 10⁻⁵) × (5.0 × 10⁶). Step 2: Multiply the coefficients: 9.2 × 5.0 = 46. Step 3: Add the exponents: 10⁻⁵ × 10⁶ = 10⁻⁵⁺⁶ = 10¹. Step 4: Combine: 46 × 10¹ = 46 × 10 = 460. Step 5: The result is already in standard form: 460 grams. The answer is 460.

6. Tane is a conservation biologist studying the population of kauri trees in a New Zealand forest. He estimates that there are 1.5 × 10^5 kauri trees in the forest. Each mature kauri tree produces approximately 7.5 × 10^3 seeds per year. However, only 1 out of every 5.0 × 10^2 seeds successfully germinates and grows into a seedling. How many new kauri seedlings are produced in the forest each year? Express your answer in scientific notation. Answer: 2.25 × 10^6 seedlings Solution: Find the total number of seeds produced per year. Multiply the number of trees by the seeds per tree: (1.5 × 10^5) × (7.5 × 10^3) = (1.5 × 7.5) × (10^5 × 10^3) = 11.25 × 10^8 = 1.125 × 10^9 seeds.
   Full step-by-step solution

   Step 1: Find the total number of seeds produced per year. Multiply the number of trees by the seeds per tree: (1.5 × 10^5) × (7.5 × 10^3) = (1.5 × 7.5) × (10^5 × 10^3) = 11.25 × 10^8 = 1.125 × 10^9 seeds. Step 2: Only 1 out of every 5.0 × 10^2 seeds germinates. This means we divide the total seeds by 5.0 × 10^2: (1.125 × 10^9) ÷ (5.0 × 10^2) = (1.125 ÷ 5.0) × (10^9 ÷ 10^2) = 0.225 × 10^7 = 2.25 × 10^6. Step 3: Check that the coefficient is between 1 and 10 (2.25 is valid) and the units are seedlings. The answer is 2.25 × 10^6 seedlings.

7. Liam's warehouse stores 7.5 × 10⁷ kilograms of grain. Each shipment removes 1.5 × 10³ kilograms. How many shipments are needed to remove all the grain? Answer: 50000 Solution: We need to divide the total grain by the amount per shipment: (7.5 × 10⁷) ÷ (1.5 × 10³). Divide the coefficients: 7.5 ÷ 1.5 = 5. Subtract the exponents: 10⁷ ÷ 10³ = 10⁷⁻³ = 10⁴.
   Full step-by-step solution

   Step 1: We need to divide the total grain by the amount per shipment: (7.5 × 10⁷) ÷ (1.5 × 10³). Step 2: Divide the coefficients: 7.5 ÷ 1.5 = 5. Step 3: Subtract the exponents: 10⁷ ÷ 10³ = 10⁷⁻³ = 10⁴. Step 4: Combine: 5 × 10⁴. Step 5: Convert to standard form: 5 × 10000 = 50000. The answer is 50000.