Scientific Operations

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Scientific Operations · Grade 7 · Worksheet 2

1. A research lab is studying bacteria growth. They observe that a certain bacteria colony doubles in size every 3 hours. If the colony starts with 1.5 × 10^4 bacteria, how many bacteria will be present after 12 hours? Express your answer in scientific notation. Answer: 2.4 × 10^5 Solution: The bacteria colony doubles every 3 hours. Initial number of bacteria = 1.5 × 10^4. Time elapsed = 12 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The bacteria colony doubles every 3 hours. Initial number of bacteria = 1.5 × 10^4. Time elapsed = 12 hours. We need the number after 12 hours. --- **Step 2: Determine the number of doubling periods** Doubling time = 3 hours. Number of doublings in 12 hours = 12 / 3 = 4 doublings. --- **Step 3: Apply the doubling formula** Final amount = Initial amount × (2^number of doublings) Final amount = (1.5 × 10^4) × (2^4) --- **Step 4: Calculate 2^4** 2^4 = 16 --- **Step 5: Multiply** First multiply 1.5 × 16 = 24 So: (1.5 × 10^4) × 16 = 24 × 10^4 --- **Step 6: Convert to proper scientific notation** 24 × 10^4 = 2.4 × 10^1 × 10^4 = 2.4 × 10^(1 + 4) = 2.4 × 10^5 --- **Step 7: Final answer** After 12 hours, the number of bacteria is **2.4 × 10^5**.

2. (6.5 × 10⁷) × (4.0 × 10³) = ? Answer: 2.6 × 10¹¹ Solution: Write the expression: (6.5 × 10⁷) × (4.0 × 10³) Separate the coefficients and powers of ten: (6.5 × 4.0) × (10⁷ × 10³) Multiply the coefficients: 6.5 × 4.0 = 26.0 Multiply the powers of ten by adding exponents: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Combine: 26.0 × 10¹⁰ Adjust to proper scientific…
   Full step-by-step solution

   Step 1: Write the expression: (6.5 × 10⁷) × (4.0 × 10³) Step 2: Separate the coefficients and powers of ten: (6.5 × 4.0) × (10⁷ × 10³) Step 3: Multiply the coefficients: 6.5 × 4.0 = 26.0 Step 4: Multiply the powers of ten by adding exponents: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Step 5: Combine: 26.0 × 10¹⁰ Step 6: Adjust to proper scientific notation: 26.0 is not between 1 and 10, so rewrite 26.0 as 2.6 × 10¹, then multiply: 2.6 × 10¹ × 10¹⁰ = 2.6 × 10¹¹ The answer is 2.6 × 10¹¹.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). What is the length of the hypotenuse of this triangle? Answer: 13 Solution: Identify the coordinates of the triangle's vertices: (0,0), (12,0), and (12,5). Calculate the length of the horizontal leg: The distance between (0,0) and (12,0) is |12 - 0| = 12 units.
   Full step-by-step solution

   Step 1: Identify the coordinates of the triangle's vertices: (0,0), (12,0), and (12,5). Step 2: Calculate the length of the horizontal leg: The distance between (0,0) and (12,0) is |12 - 0| = 12 units. Step 3: Calculate the length of the vertical leg: The distance between (12,0) and (12,5) is |5 - 0| = 5 units. Step 4: Apply the Pythagorean theorem: hypotenuse² = leg1² + leg2² = 12² + 5² = 144 + 25 = 169. Step 5: Find the hypotenuse: sqrt(169) = 13. The length of the hypotenuse is 13 units.

4. 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 the 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 (the coefficients) first. 2.5 × 8 × 4 = 20 × 4 = 80 Step 4: Multiply the powers of 10 using the rule: 10^a × 10^b × 10^c = 10^(a+b+c). 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 Step 7: Multiply 8.0 × 10¹ by 10⁶. 8.0 × 10¹ × 10⁶ = 8.0 × 10^(1+6) = 8.0 × 10⁷ Step 8: Write the final answer in scientific notation. Volume = 8 × 10⁷ cm³ This is the volume of the rectangular prism in scientific notation.

5. (9.2 × 10⁷) × (4.5 × 10³) = ? Answer: 4.14 × 10¹¹ Solution: Write the expression: (9.2 × 10⁷) × (4.5 × 10³) Multiply the coefficients: 9.2 × 4.5 = 41.4 Multiply the powers of 10: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Combine: 41.4 × 10¹⁰ Adjust to proper scientific notation: 41.4 = 4.14 × 10¹, so 41.4 × 10¹⁰ = 4.14 × 10¹ × 10¹⁰ = 4.14 × 10¹¹ The answer is 4.14 × 10¹¹.
   Full step-by-step solution

   Step 1: Write the expression: (9.2 × 10⁷) × (4.5 × 10³) Step 2: Multiply the coefficients: 9.2 × 4.5 = 41.4 Step 3: Multiply the powers of 10: 10⁷ × 10³ = 10^(7+3) = 10¹⁰ Step 4: Combine: 41.4 × 10¹⁰ Step 5: Adjust to proper scientific notation: 41.4 = 4.14 × 10¹, so 41.4 × 10¹⁰ = 4.14 × 10¹ × 10¹⁰ = 4.14 × 10¹¹ The answer is 4.14 × 10¹¹.

6. A rectangular prism is drawn with dimensions 12 cm by 8 cm by 15 cm. If you were to calculate the volume of this prism, what would be your result in cubic centimeters? Answer: 1440 Solution: We have a rectangular prism with length 12 cm, width 8 cm, and height 15 cm. The volume of a rectangular prism is found by multiplying its length, width, and height. Write the volume formula.
   Full step-by-step solution

   Step 1: Understand the problem. We have a rectangular prism with length 12 cm, width 8 cm, and height 15 cm. The volume of a rectangular prism is found by multiplying its length, width, and height. Step 2: Write the volume formula. Volume = length × width × height Step 3: Substitute the given values into the formula. Volume = 12 × 8 × 15 Step 4: Multiply the first two numbers. 12 × 8 = 96 Step 5: Multiply the result by the third number. 96 × 15 = 1440 Step 6: State the final answer with units. The volume is 1440 cubic centimeters. Final Answer: 1440

7. 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.