Scientific Operations

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Answer Key & Explanations

Scientific Operations · Grade 7 · Worksheet 1

1. (6.4 × 10⁸) ÷ (1.6 × 10²) = ? Answer: 4000000 Solution: Write the expression: (6.4 × 10⁸) ÷ (1.6 × 10²) Divide the coefficients: 6.4 ÷ 1.6 = 4 Divide the powers of ten: 10⁸ ÷ 10² = 10^(8-2) = 10⁶ Combine the results: 4 × 10⁶ Convert to standard form: 4 × 1,000,000 = 4,000,000 The answer is 4000000.
   Full step-by-step solution

   Step 1: Write the expression: (6.4 × 10⁸) ÷ (1.6 × 10²) Step 2: Divide the coefficients: 6.4 ÷ 1.6 = 4 Step 3: Divide the powers of ten: 10⁸ ÷ 10² = 10^(8-2) = 10⁶ Step 4: Combine the results: 4 × 10⁶ Step 5: Convert to standard form: 4 × 1,000,000 = 4,000,000 The answer is 4000000.

2. A scientist is studying the distance between Earth and a newly discovered exoplanet. The distance is approximately 4.2 × 10^13 kilometers. A space probe travels at a constant speed of 3.5 × 10^4 kilometers per hour. How many hours will it take the probe to reach the exoplanet? Express your answer in scientific notation. Answer: 1.2 × 10^9 Solution: When solving problems involving rates like speed, time equals distance divided by speed.
   Full step-by-step solution

   When solving problems involving rates like speed, time equals distance divided by speed. With scientific notation, you handle the coefficients and exponents separately, making calculations with very large or very small numbers more manageable.

3. A cylindrical water tank is drawn with a height of 2.5 × 10² cm and a diameter of 8.0 × 10¹ cm. Using π = 3.14, what is the volume of the tank in cubic centimeters? Express your final answer in scientific notation. Answer: 1.256 × 10^6 Solution: Find the radius of the cylinder. The diameter is 8.0 × 10¹ cm, so the radius is half of that: (8.0 × 10¹) ÷ 2 = 4.0 × 10¹ cm. Calculate the area of the circular base using the formula A = πr².
   Full step-by-step solution

   Step 1: Find the radius of the cylinder. The diameter is 8.0 × 10¹ cm, so the radius is half of that: (8.0 × 10¹) ÷ 2 = 4.0 × 10¹ cm. Step 2: Calculate the area of the circular base using the formula A = πr². A = 3.14 × (4.0 × 10¹)² = 3.14 × (1.6 × 10³) = 3.14 × 1600 = 5024 cm². Step 3: Calculate the volume using the formula V = A × h. V = 5024 × (2.5 × 10²) = 5024 × 250 = 1,256,000 cm³. Step 4: Convert 1,256,000 to scientific notation: 1.256 × 10^6. The volume of the tank is 1.256 × 10^6 cm³.

4. A rectangular solar panel has a length of 2.4 × 10^4 millimeters and a width of 8.5 × 10^3 millimeters. What is the area of the solar panel in square millimeters? Express your answer in scientific notation. Answer: 2.04×10^8 Solution: Write down the formula for the area of a rectangle. Area = length × width Substitute the given values into the formula.
   Full step-by-step solution

   Step 1: Write down the formula for the area of a rectangle. Area = length × width Step 2: Substitute the given values into the formula. Length = 2.4 × 10^4 mm Width = 8.5 × 10^3 mm Area = (2.4 × 10^4) × (8.5 × 10^3) Step 3: Multiply the decimal parts (coefficients) first. 2.4 × 8.5 = 20.4 Step 4: Multiply the powers of 10. 10^4 × 10^3 = 10^(4+3) = 10^7 Step 5: Combine the results from Steps 3 and 4. Area = 20.4 × 10^7 Step 6: Adjust to proper scientific notation. In scientific notation, the coefficient must be between 1 and 10. 20.4 = 2.04 × 10 So, 20.4 × 10^7 = 2.04 × 10 × 10^7 = 2.04 × 10^(1+7) = 2.04 × 10^8 Step 7: Final answer. Area = 2.04 × 10^8 square millimeters

5. A research lab is studying bacterial growth. They observe that a particular bacteria colony doubles in size every 3 hours. If the colony starts with 8,000 bacteria, how many bacteria will there be after 12 hours? Express your answer in scientific notation. Answer: 1.28 × 10^5 Solution: The bacteria colony doubles in size every 3 hours. Initial number of bacteria = 8,000. We want the number after 12 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The bacteria colony doubles in size every 3 hours. Initial number of bacteria = 8,000. We want the number after 12 hours. --- **Step 2: Determine how many doubling periods occur in 12 hours** Doubling time = 3 hours. Number of periods = Total time / Doubling time = 12 / 3 = 4 periods. So the colony doubles 4 times. --- **Step 3: Apply the doubling formula** Final amount = Initial amount × (2 ^ number of periods) = 8,000 × (2^4) = 8,000 × 16 --- **Step 4: Calculate** 8,000 × 16: 8,000 × 10 = 80,000 8,000 × 6 = 48,000 Add: 80,000 + 48,000 = 128,000 So after 12 hours, there are 128,000 bacteria. --- **Step 5: Convert to scientific notation** 128,000 = 1.28 × 100,000 100,000 = 10^5 So 128,000 = 1.28 × 10^5 --- **Final Answer:** 1.28 × 10^5

6. A scientist is studying the growth of bacteria in two different petri dishes. Dish A starts with 3.2 × 10^4 bacteria and doubles every 4 hours. Dish B starts with 1.5 × 10^5 bacteria and doubles every 6 hours. After 12 hours, which dish will have more bacteria and by how much? Express your answer in scientific notation. Answer: Dish B has 1.2 × 10^5 more bacteria Solution: Calculate bacteria in Dish A after 12 hours - Starting bacteria: 3.2 × 10^4 - Doubling time: 4 hours - Number of doublings in 12 hours: 12 ÷ 4 = 3 doublings - After 3 doublings: 3.2 × 10^4 × 2^3 = 3.2 × 10^4 × 8 = 25.6 × 10^4 = 2.56 × 10^5 Calculate bacteria in Dish B after 12 hours - Starting…
   Full step-by-step solution

   Step 1: Calculate bacteria in Dish A after 12 hours - Starting bacteria: 3.2 × 10^4 - Doubling time: 4 hours - Number of doublings in 12 hours: 12 ÷ 4 = 3 doublings - After 3 doublings: 3.2 × 10^4 × 2^3 = 3.2 × 10^4 × 8 = 25.6 × 10^4 = 2.56 × 10^5 Step 2: Calculate bacteria in Dish B after 12 hours - Starting bacteria: 1.5 × 10^5 - Doubling time: 6 hours - Number of doublings in 12 hours: 12 ÷ 6 = 2 doublings - After 2 doublings: 1.5 × 10^5 × 2^2 = 1.5 × 10^5 × 4 = 6.0 × 10^5 Step 3: Compare the populations - Dish A: 2.56 × 10^5 - Dish B: 6.0 × 10^5 - Difference: 6.0 × 10^5 - 2.56 × 10^5 = 3.44 × 10^5 = 3.44 × 10^5 - Since 3.44 × 10^5 = 3.44 × 10^5, and Dish B has more bacteria Step 4: Express the difference properly 6.0 × 10^5 - 2.56 × 10^5 = 3.44 × 10^5 Dish B has 3.44 × 10^5 more bacteria than Dish A.

7. A factory produces 2.5 × 10⁴ computer chips per day. If each chip requires 8 × 10⁻⁵ grams of gold plating, how many grams of gold are used in total each day? Express your answer in scientific notation. Answer: 2 Solution: Identify the quantities given. - Number of chips per day = 2.5 × 10⁴ - Gold per chip = 8 × 10⁻⁵ grams Multiply the two numbers to find total gold used.
   Full step-by-step solution

   Step 1: Identify the quantities given. - Number of chips per day = 2.5 × 10⁴ - Gold per chip = 8 × 10⁻⁵ grams Step 2: Multiply the two numbers to find total gold used. Total gold = (2.5 × 10⁴) × (8 × 10⁻⁵) Step 3: Multiply the decimal parts first. 2.5 × 8 = 20 Step 4: Multiply the powers of 10. 10⁴ × 10⁻⁵ = 10^(4 + (-5)) = 10⁻¹ Step 5: Combine the results. 20 × 10⁻¹ Step 6: Simplify 20 × 10⁻¹. 20 × 10⁻¹ = 20 × (1/10) = 20/10 = 2 Step 7: Express 2 in scientific notation. 2 = 2 × 10⁰ Final answer: 2 × 10⁰ grams of gold per day.

8. Liam is an astronomer studying the Andromeda Galaxy. He knows the galaxy is approximately 2.5 × 10^6 light-years away from Earth. A light-year is about 9.46 × 10^12 kilometers. How many kilometers away is the Andromeda Galaxy? Express your answer in scientific notation. Answer: 2.365 × 10^19 Solution: Scientific notation is used to represent very large or very small numbers. When multiplying numbers in scientific notation, you multiply the decimal parts and add the exponents of the 10.
   Full step-by-step solution

   Scientific notation is used to represent very large or very small numbers. When multiplying numbers in scientific notation, you multiply the decimal parts and add the exponents of the 10. This method helps simplify calculations with extremely large numbers like those used in astronomy.