Scientific Operations

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Answer Key & Explanations

Scientific Operations · Grade 7 · Worksheet 3

1. (7.5 × 10⁹) ÷ (2.5 × 10³) = ? Answer: 3000000 Solution: Write the expression: (7.5 × 10⁹) ÷ (2.5 × 10³) Divide the coefficients: 7.5 ÷ 2.5 = 3 Subtract the exponents: 10⁹ ÷ 10³ = 10^(9-3) = 10⁶ Combine the results: 3 × 10⁶ Convert to standard form: 3 × 1,000,000 = 3,000,000 The answer is 3000000.
   Full step-by-step solution

   Step 1: Write the expression: (7.5 × 10⁹) ÷ (2.5 × 10³) Step 2: Divide the coefficients: 7.5 ÷ 2.5 = 3 Step 3: Subtract the exponents: 10⁹ ÷ 10³ = 10^(9-3) = 10⁶ Step 4: Combine the results: 3 × 10⁶ Step 5: Convert to standard form: 3 × 1,000,000 = 3,000,000 The answer is 3000000.

2. A scientist is studying the growth of bacteria in two different petri dishes. Dish A starts with 3.2 × 10^5 bacteria and doubles every 4 hours. Dish B starts with 1.8 × 10^6 bacteria and triples 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 1.26 × 10^7 more bacteria Solution: When dealing with exponential growth over multiple time periods, you need to determine how many complete growth cycles occur. The total growth is calculated by raising the growth factor to the number of cycles.
   Full step-by-step solution

   When dealing with exponential growth over multiple time periods, you need to determine how many complete growth cycles occur. The total growth is calculated by raising the growth factor to the number of cycles. For example, if something doubles every 2 hours over 8 hours, it would double 4 times, so the growth factor would be 2^4 = 16 times the original amount.

3. (3.4 × 10⁵) × (2.5 × 10³) = ? Answer: 8.5 × 10⁸ Solution: Separate the decimal parts and the powers of 10. We have: (3.4 × 2.5) × (10⁵ × 10³) Multiply the decimal parts. 3.4 × 2.5 = ?
   Full step-by-step solution

   Let's multiply (3.4 × 10⁵) × (2.5 × 10³) step by step. Step 1: Separate the decimal parts and the powers of 10. We have: (3.4 × 2.5) × (10⁵ × 10³) Step 2: Multiply the decimal parts. 3.4 × 2.5 = ? First, 3 × 2.5 = 7.5 Then, 0.4 × 2.5 = 1.0 Add them: 7.5 + 1.0 = 8.5 So, 3.4 × 2.5 = 8.5 Step 3: Multiply the powers of 10. 10⁵ × 10³ = 10^(5 + 3) = 10⁸ (When multiplying powers with the same base, add the exponents.) Step 4: Combine the results. 8.5 × 10⁸ Step 5: Check if the decimal part is in correct scientific notation. 8.5 is between 1 and 10, so it's already in proper scientific notation. Final answer: 8.5 × 10⁸

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). What is the length of the hypotenuse of this triangle? Answer: 10 Solution: Identify the side lengths from the coordinates. The points (0,0) and (6,0) have the same y-coordinate, so the horizontal leg length is |6 - 0| = 6 units.
   Full step-by-step solution

   Step 1: Identify the side lengths from the coordinates. The points (0,0) and (6,0) have the same y-coordinate, so the horizontal leg length is |6 - 0| = 6 units. The points (0,0) and (0,8) have the same x-coordinate, so the vertical leg length is |8 - 0| = 8 units. Step 2: Apply the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. Step 3: Substitute the values: 6² + 8² = c² Step 4: Calculate: 36 + 64 = 100 Step 5: Find c by taking the square root: c = sqrt(100) = 10 Step 6: The hypotenuse is 10 units long.

5. A cylindrical water tank is drawn with a height of 2.5 × 10² cm and a diameter of 8 × 10¹ cm. Using π = 3.14, what is the volume of the cylinder in cubic centimeters? Express your answer in scientific notation. Answer: 1.256 × 10^6 Solution: Find the radius of the cylinder. The diameter is 8 × 10¹ cm, so the radius is half of that: (8 × 10¹)/2 = 4 × 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 × 10¹ cm, so the radius is half of that: (8 × 10¹)/2 = 4 × 10¹ cm. Step 2: Calculate the area of the circular base using the formula A = πr². A = 3.14 × (4 × 10¹)² = 3.14 × (16 × 10²) = 3.14 × 1.6 × 10³ = 5.024 × 10³ cm². Step 3: Calculate the volume using the formula V = base area × height. V = (5.024 × 10³) × (2.5 × 10²) = 5.024 × 2.5 × 10^(3+2) = 12.56 × 10⁵ = 1.256 × 10⁶ cm³. The volume of the cylinder is 1.256 × 10⁶ cubic centimeters.

6. A research lab is studying bacteria growth. They observe that a particular colony doubles in size every 3 hours. If the colony starts with 1.5 × 10^4 bacteria, how many bacteria will there be after 9 hours? Express your answer in scientific notation. Answer: 1.2 × 10^5 Solution: The colony starts with \( 1.5 \times 10^4 \) bacteria. It doubles every 3 hours. We want the number after 9 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The colony starts with \( 1.5 \times 10^4 \) bacteria. It doubles every 3 hours. We want the number after 9 hours. --- **Step 2: Determine how many doubling periods occur in 9 hours** Doubling time = 3 hours Total time = 9 hours Number of doubling periods = \( 9 / 3 = 3 \) So the colony doubles 3 times. --- **Step 3: Apply the doubling** Start: \( N_0 = 1.5 \times 10^4 \) After 1 doubling: \( N_1 = 1.5 \times 10^4 \times 2 \) After 2 doublings: \( N_2 = 1.5 \times 10^4 \times 2^2 \) After 3 doublings: \( N_3 = 1.5 \times 10^4 \times 2^3 \) --- **Step 4: Calculate \( 2^3 \)** \( 2^3 = 8 \) So \( N_3 = 1.5 \times 10^4 \times 8 \) --- **Step 5: Multiply** First multiply 1.5 by 8: \( 1.5 \times 8 = 12 \) So \( N_3 = 12 \times 10^4 \) --- **Step 6: Convert to proper scientific notation** \( 12 \times 10^4 = (1.2 \times 10^1) \times 10^4 \) \( = 1.2 \times 10^{1+4} \) \( = 1.2 \times 10^5 \) --- **Final Answer:** \( 1.2 \times 10^5 \)

7. A research lab is studying bacteria growth. They observe that a particular colony doubles in size every 3 hours. If the colony starts with 1.5 × 10³ bacteria, how many bacteria will there be after 12 hours? Express your answer in scientific notation. Answer: 2.4 × 10⁴ Solution: The colony doubles every 3 hours.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Understand the doubling time and total time** The colony doubles every 3 hours. Starting bacteria count = 1.5 × 10³ Total time = 12 hours **Step 2: Determine how many doubling periods fit into 12 hours** Doubling period = 3 hours Number of doublings = Total time / Doubling time = 12 / 3 = 4 So the colony doubles 4 times. **Step 3: Formula for final amount** Final amount = Initial amount × (2^(number of doublings)) Final amount = (1.5 × 10³) × (2⁴) **Step 4: Compute 2⁴** 2⁴ = 16 **Step 5: Multiply initial amount by 16** 1.5 × 10³ × 16 First multiply 1.5 × 16 = 24 So we have 24 × 10³ **Step 6: Convert to proper scientific notation** 24 × 10³ = 2.4 × 10¹ × 10³ = 2.4 × 10^(1+3) = 2.4 × 10⁴ **Step 7: Final answer** After 12 hours, the colony will have 2.4 × 10⁴ bacteria.

8. (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.