Scientific Operations

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Scientific Operations · Grade 7 · Worksheet 1

1. 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

2. (8.8 × 10⁷) × (9.5 × 10³) = ? Answer: 8.36 × 10¹¹ Solution: Separate the coefficients and powers of 10: (8.8 × 9.5) × (10⁷ × 10³) Multiply the coefficients: 8.8 × 9.5 = 83.6 Multiply the powers of 10: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Combine: 83.6 × 10¹⁰ Convert to proper scientific notation: 83.6 = 8.36 × 10¹, so 83.6 × 10¹⁰ = 8.36 × 10¹ × 10¹⁰ = 8.36 × 10¹¹…
   Full step-by-step solution

   Step 1: Separate the coefficients and powers of 10: (8.8 × 9.5) × (10⁷ × 10³) Step 2: Multiply the coefficients: 8.8 × 9.5 = 83.6 Step 3: Multiply the powers of 10: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Step 4: Combine: 83.6 × 10¹⁰ Step 5: Convert to proper scientific notation: 83.6 = 8.36 × 10¹, so 83.6 × 10¹⁰ = 8.36 × 10¹ × 10¹⁰ = 8.36 × 10¹¹ The answer is 8.36 × 10¹¹.

3. Mere is drawing a scale model of a rectangular park. On her diagram, the park has a length of 3.5 × 10⁻¹ meters and a width of 2.0 × 10⁻¹ meters. In the actual park, the length is 1.4 × 10³ meters. If the park is a scaled rectangle, what is the area of the actual park in square meters? Express your answer in scientific notation. Answer: 1.12 × 10⁶ Solution: Find the scale factor from the diagram to the actual park. Scale factor = Actual length ÷ Diagram length Scale factor = (1.4 × 10³) ÷ (3.5 × 10⁻¹) Scale factor = (1.4 ÷ 3.5) × (10³ ÷ 10⁻¹) Scale factor = 0.4 × 10^(3 - (-1)) Scale factor = 0.4 × 10⁴ Scale factor = 4.0 × 10³ Find the actual width.
   Full step-by-step solution

   Step 1: Find the scale factor from the diagram to the actual park. Scale factor = Actual length ÷ Diagram length Scale factor = (1.4 × 10³) ÷ (3.5 × 10⁻¹) Scale factor = (1.4 ÷ 3.5) × (10³ ÷ 10⁻¹) Scale factor = 0.4 × 10^(3 - (-1)) Scale factor = 0.4 × 10⁴ Scale factor = 4.0 × 10³ Step 2: Find the actual width. Actual width = Diagram width × Scale factor Actual width = (2.0 × 10⁻¹) × (4.0 × 10³) Actual width = (2.0 × 4.0) × (10⁻¹ × 10³) Actual width = 8.0 × 10² meters Step 3: Calculate the area of the actual park. Area = Length × Width Area = (1.4 × 10³) × (8.0 × 10²) Area = (1.4 × 8.0) × (10³ × 10²) Area = 11.2 × 10⁵ Area = 1.12 × 10⁶ square meters The area of the actual park is 1.12 × 10⁶ square meters.

4. 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.

5. A research lab is studying bacteria growth. They observe that a particular bacteria 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 bacteria double every 3 hours. Initial number of bacteria: \( 1.5 \times 10^3 \). Determine how many doubling periods in 12 hours Doubling time = 3 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the doubling time** The bacteria double every 3 hours. Initial number of bacteria: \( 1.5 \times 10^3 \). --- **Step 2: Determine how many doubling periods in 12 hours** Doubling time = 3 hours. Total time = 12 hours. Number of doubling periods = \( \frac{12}{3} = 4 \). So the colony doubles 4 times. --- **Step 3: Apply the doubling formula** After \( n \) doublings: Final amount = Initial amount × \( 2^n \). Here: Initial = \( 1.5 \times 10^3 \) \( n = 4 \) Final = \( 1.5 \times 10^3 \times 2^4 \). --- **Step 4: Calculate \( 2^4 \)** \( 2^4 = 16 \). So: Final = \( 1.5 \times 10^3 \times 16 \). --- **Step 5: Multiply numbers** First multiply \( 1.5 \times 16 \): \( 1.5 \times 16 = 24 \). So: Final = \( 24 \times 10^3 \). --- **Step 6: Convert to proper scientific notation** \( 24 \times 10^3 = 2.4 \times 10^1 \times 10^3 \). \( 10^1 \times 10^3 = 10^{4} \). So: Final = \( 2.4 \times 10^4 \). --- **Step 7: Final answer** After 12 hours, the number of bacteria is \( 2.4 \times 10^4 \).

6. Hana draws a rectangular solar panel on a grid. The length of the panel is represented by 4.8 × 10⁴ mm and the width is represented by 2.5 × 10³ mm. What is the area of the solar panel in square millimeters? Express your answer in scientific notation. Answer: 1.2 × 10⁸ Solution: Write the formula for area of a rectangle: Area = length × width. Substitute the given values: Area = (4.8 × 10⁴) × (2.5 × 10³). Multiply the coefficients: 4.8 × 2.5 = 12.
   Full step-by-step solution

   Step 1: Write the formula for area of a rectangle: Area = length × width. Step 2: Substitute the given values: Area = (4.8 × 10⁴) × (2.5 × 10³). Step 3: Multiply the coefficients: 4.8 × 2.5 = 12. Step 4: Multiply the powers of 10 by adding the exponents: 10⁴ × 10³ = 10⁽⁴⁺³⁾ = 10⁷. Step 5: Combine the results: 12 × 10⁷. Step 6: Adjust to proper scientific notation (coefficient between 1 and 10): 12 × 10⁷ = 1.2 × 10¹ × 10⁷ = 1.2 × 10⁸. The area of the solar panel is 1.2 × 10⁸ square millimeters.

7. (6.1 × 10⁶) × (1.6 × 10⁴) = ? Answer: 9.76 × 10¹⁰ Solution: Write the expression: (6.1 × 10⁶) × (1.6 × 10⁴) Multiply the coefficients: 6.1 × 1.6 = 9.76 Multiply the powers of 10: 10⁶ × 10⁴ = 10^(6+4) = 10¹⁰ Combine the results: 9.76 × 10¹⁰ Check if the coefficient is between 1 and 10: 9.76 is between 1 and 10, so the answer is already in proper…
   Full step-by-step solution

   Step 1: Write the expression: (6.1 × 10⁶) × (1.6 × 10⁴) Step 2: Multiply the coefficients: 6.1 × 1.6 = 9.76 Step 3: Multiply the powers of 10: 10⁶ × 10⁴ = 10^(6+4) = 10¹⁰ Step 4: Combine the results: 9.76 × 10¹⁰ Step 5: Check if the coefficient is between 1 and 10: 9.76 is between 1 and 10, so the answer is already in proper scientific notation. The answer is 9.76 × 10¹⁰.

8. (9.1 × 10⁷) ÷ (1.3 × 10³) = ? Answer: 70000 Solution: Write the expression: (9.1 × 10⁷) ÷ (1.3 × 10³) Divide the coefficients: 9.1 ÷ 1.3 = 7 Divide the powers of 10: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Combine the results: 7 × 10⁴ Convert to standard form: 7 × 10000 = 70000 The answer is 70000.
   Full step-by-step solution

   Step 1: Write the expression: (9.1 × 10⁷) ÷ (1.3 × 10³) Step 2: Divide the coefficients: 9.1 ÷ 1.3 = 7 Step 3: Divide the powers of 10: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Step 4: Combine the results: 7 × 10⁴ Step 5: Convert to standard form: 7 × 10000 = 70000 The answer is 70000.