Scientific Operations

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Scientific Operations · Grade 7 · Worksheet 2

1. A research lab is studying bacterial growth. A scientist observes that a particular bacterial colony doubles in size every 3 hours. If the colony starts with 5,000 bacteria, how many bacteria will there be after 15 hours? Answer: 160000 Solution: The colony starts with 5,000 bacteria. It doubles every 3 hours. We want the number of bacteria after 15 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The colony starts with 5,000 bacteria. It doubles every 3 hours. We want the number of bacteria after 15 hours. --- **Step 2: Determine the number of doubling periods** Doubling time = 3 hours. Total time = 15 hours. Number of doublings = Total time / Doubling time = 15 / 3 = 5 doublings. --- **Step 3: Apply the exponential growth formula** Initial bacteria count = 5,000. After n doublings: Final count = Initial × (2^n) Here n = 5. Final count = 5,000 × (2^5). --- **Step 4: Calculate step-by-step** 2^5 = 2 × 2 × 2 × 2 × 2 = 32. So Final count = 5,000 × 32. --- **Step 5: Multiply** 5,000 × 32 = 5,000 × (30 + 2) = 5,000 × 30 + 5,000 × 2 = 150,000 + 10,000 = 160,000. --- **Step 6: Conclusion** After 15 hours, the colony will have 160,000 bacteria. --- **Final answer:** 160000

2. (8.8 × 10⁷) ÷ (2.2 × 10³) = ? Answer: 40000 Solution: Write the expression: (8.8 × 10⁷) ÷ (2.2 × 10³) Divide the coefficients: 8.8 ÷ 2.2 = 4 Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Combine: 4 × 10⁴ Convert to standard form: 4 × 10,000 = 40,000 The answer is 40000.
   Full step-by-step solution

   Step 1: Write the expression: (8.8 × 10⁷) ÷ (2.2 × 10³) Step 2: Divide the coefficients: 8.8 ÷ 2.2 = 4 Step 3: Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Step 4: Combine: 4 × 10⁴ Step 5: Convert to standard form: 4 × 10,000 = 40,000 The answer is 40000.

3. A rectangular prism has a length of 2.5 × 10³ cm, a width of 8 × 10² cm, and a height of 4 × 10¹ cm. What is the volume of this prism in scientific notation? Answer: 8 × 10⁷ cm³ Solution: Write down the formula for the volume of a rectangular prism. Volume = length × width × height Substitute the given values into the formula.
   Full step-by-step solution

   Step 1: Write down the formula for the volume of a rectangular prism. Volume = length × width × height Step 2: Substitute the given values into the formula. Length = 2.5 × 10³ cm Width = 8 × 10² cm Height = 4 × 10¹ cm Volume = (2.5 × 10³) × (8 × 10²) × (4 × 10¹) Step 3: Multiply the decimal numbers (coefficients) together. 2.5 × 8 = 20 20 × 4 = 80 So the coefficient part is 80. Step 4: Multiply the powers of 10 together. 10³ × 10² × 10¹ = 10^(3 + 2 + 1) = 10⁶ Step 5: Combine the results from Step 3 and Step 4. Volume = 80 × 10⁶ cm³ Step 6: Convert 80 into scientific notation. 80 = 8.0 × 10¹ So: Volume = (8.0 × 10¹) × 10⁶ cm³ Step 7: Multiply the powers of 10. 10¹ × 10⁶ = 10^(1 + 6) = 10⁷ Step 8: Final result. Volume = 8 × 10⁷ cm³ This is the volume of the rectangular prism in scientific notation.

4. (9.6 × 10⁸) ÷ (1.2 × 10⁴) = ? Answer: 80000 Solution: Write the expression: (9.6 × 10⁸) ÷ (1.2 × 10⁴) Divide the coefficients: 9.6 ÷ 1.2 = 8 Subtract the exponents: 10⁸ ÷ 10⁴ = 10^(8-4) = 10⁴ Combine the results: 8 × 10⁴ Convert to standard form: 8 × 10,000 = 80,000 The answer is 80000.
   Full step-by-step solution

   Step 1: Write the expression: (9.6 × 10⁸) ÷ (1.2 × 10⁴) Step 2: Divide the coefficients: 9.6 ÷ 1.2 = 8 Step 3: Subtract the exponents: 10⁸ ÷ 10⁴ = 10^(8-4) = 10⁴ Step 4: Combine the results: 8 × 10⁴ Step 5: Convert to standard form: 8 × 10,000 = 80,000 The answer is 80000.

5. (6.1 × 10⁶) × (1.6 × 10³) = ? Answer: 9.76 × 10⁹ Solution: Write the expression: (6.1 × 10⁶) × (1.6 × 10³) Separate the coefficients and powers of 10: (6.1 × 1.6) × (10⁶ × 10³) Multiply the coefficients: 6.1 × 1.6 = 9.76 Multiply the powers of 10: 10⁶ × 10³ = 10^(6+3) = 10⁹ Combine: 9.76 × 10⁹ Check if coefficient is between 1 and 10: 9.76 is between 1…
   Full step-by-step solution

   Step 1: Write the expression: (6.1 × 10⁶) × (1.6 × 10³) Step 2: Separate the coefficients and powers of 10: (6.1 × 1.6) × (10⁶ × 10³) Step 3: Multiply the coefficients: 6.1 × 1.6 = 9.76 Step 4: Multiply the powers of 10: 10⁶ × 10³ = 10^(6+3) = 10⁹ Step 5: Combine: 9.76 × 10⁹ Step 6: Check if coefficient is between 1 and 10: 9.76 is between 1 and 10, so it is already in proper scientific notation. The answer is 9.76 × 10⁹.

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 × 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.

7. (8.5 × 10⁷) ÷ (5 × 10³) = ? Answer: 17000 Solution: Write the expression: (8.5 × 10⁷) ÷ (5 × 10³) Divide the coefficients: 8.5 ÷ 5 = 1.7 Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Combine: 1.7 × 10⁴ Convert to standard form: 1.7 × 10000 = 17000 The answer is 17000.
   Full step-by-step solution

   Step 1: Write the expression: (8.5 × 10⁷) ÷ (5 × 10³) Step 2: Divide the coefficients: 8.5 ÷ 5 = 1.7 Step 3: Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Step 4: Combine: 1.7 × 10⁴ Step 5: Convert to standard form: 1.7 × 10000 = 17000 The answer is 17000.

8. A research lab is studying bacterial growth. They observe that a particular strain of bacteria doubles its population every 3 hours. If the lab starts with 1.5 × 10³ bacteria in a petri dish, how many bacteria will be present after 9 hours? Express your answer in scientific notation. Answer: 1.2 × 10⁴ Solution: The bacteria double every 3 hours. Initial population = 1.5 × 10³ Time elapsed = 9 hours Doubling time = 3 hours Number of doublings in 9 hours = 9 / 3 = 3 So the population doubles 3 times.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** The bacteria double every 3 hours. Initial population = 1.5 × 10³ Time elapsed = 9 hours --- **Step 2: Determine the number of doubling periods** Doubling time = 3 hours Number of doublings in 9 hours = 9 / 3 = 3 So the population doubles 3 times. --- **Step 3: Apply the doubling formula** Final population = Initial population × (2^(number of doublings)) Final population = (1.5 × 10³) × (2³) 2³ = 8 So: Final population = 1.5 × 10³ × 8 --- **Step 4: Multiply the numbers** 1.5 × 8 = 12 So: 12 × 10³ --- **Step 5: Convert to proper scientific notation** 12 × 10³ = 1.2 × 10¹ × 10³ = 1.2 × 10^(1+3) = 1.2 × 10⁴ --- **Step 6: Final answer** After 9 hours, there will be **1.2 × 10⁴** bacteria. --- **Answer:** 1.2 × 10⁴

9. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If the garden is enlarged using a scale factor of 3 with the origin (0, 0) as the center of dilation, what are the coordinates of the new vertices? Answer: (6, 3), (24, 3), (24, 15), (6, 15) Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Scale factor = 3 Center of dilation = (0, 0) When the center of dilation is the origin (0, 0), the rule is: New point = (scale factor × old x, scale factor × old y) (x', y') = (k × x, k × y) Here k = 3.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Scale factor = 3 Center of dilation = (0, 0) --- **Step 2: Recall the dilation formula** When the center of dilation is the origin (0, 0), the rule is: New point = (scale factor × old x, scale factor × old y) So if scale factor is k, then (x', y') = (k × x, k × y) Here k = 3. --- **Step 3: Apply to each vertex** Vertex A = (2, 1) A' = (3 × 2, 3 × 1) = (6, 3) Vertex B = (8, 1) B' = (3 × 8, 3 × 1) = (24, 3) Vertex C = (8, 5) C' = (3 × 8, 3 × 5) = (24, 15) Vertex D = (2, 5) D' = (3 × 2, 3 × 5) = (6, 15) --- **Step 4: Write the final coordinates in order** The new vertices are: (6, 3), (24, 3), (24, 15), (6, 15) --- **Step 5: Check** Original rectangle: width = 8 - 2 = 6, height = 5 - 1 = 4. After dilation: width = 24 - 6 = 18, height = 15 - 3 = 12. 18/6 = 3, 12/4 = 3, so scale factor 3 is correct. Also, distances from origin are multiplied by 3: e.g., distance from (0,0) to (2,1) is sqrt(4+1) = sqrt(5), and from (0,0) to (6,3) is sqrt(36+9) = sqrt(45) = 3 × sqrt(5). Correct. --- **Final answer:** (6, 3), (24, 3), (24, 15), (6, 15)