Scientific Notation

lessonbunny.com

Answer Key & Explanations

Scientific Notation · Grade 7 · Worksheet 3

1. The Andromeda Galaxy is approximately 2.5 million light-years away from Earth. If one light-year equals 9.46 × 10^12 kilometers, what is the distance to Andromeda in kilometers? Express your answer in scientific notation. Answer: 2.365 × 10^19 Solution: Convert 2.5 million to standard form: 2,500,000 = 2.5 × 10^6 Multiply the distance in light-years by kilometers per light-year: (2.5 × 10^6) × (9.46 × 10^12) Multiply the coefficients: 2.5 × 9.46 = 23.65 Add the exponents: 10^(6+12) = 10^18 Combine: 23.65 × 10^18 Convert to proper scientific…
   Full step-by-step solution

   Step 1: Convert 2.5 million to standard form: 2,500,000 = 2.5 × 10^6 Step 2: Multiply the distance in light-years by kilometers per light-year: (2.5 × 10^6) × (9.46 × 10^12) Step 3: Multiply the coefficients: 2.5 × 9.46 = 23.65 Step 4: Add the exponents: 10^(6+12) = 10^18 Step 5: Combine: 23.65 × 10^18 Step 6: Convert to proper scientific notation: 2.365 × 10^19 Step 7: The distance to Andromeda Galaxy is 2.365 × 10^19 kilometers.

2. (3.6 × 10⁴) × (2.5 × 10³) = ? Answer: 90000000 Solution: Multiply the coefficients: 3.6 × 2.5 = 9.0 Add the exponents: 4 + 3 = 7 Combine the results: 9.0 × 10⁷ Convert to standard form: 9.0 × 10⁷ = 90,000,000 The answer is 90000000.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 3.6 × 2.5 = 9.0 Step 2: Add the exponents: 4 + 3 = 7 Step 3: Combine the results: 9.0 × 10⁷ Step 4: Convert to standard form: 9.0 × 10⁷ = 90,000,000 The answer is 90000000.

3. A scientist is studying bacteria growth and finds that a particular colony doubles every hour. After 24 hours, the colony has reached a population of 16,777,216 bacteria. Express this population in scientific notation. Answer: 1.6777216e7 Solution: The problem says the population after 24 hours is 16,777,216 bacteria. We need to express this in scientific notation. \( a \times 10^n \) where \( 1 \leq a < 10 \) and \( n \) is an integer.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the given population number** The problem says the population after 24 hours is 16,777,216 bacteria. We need to express this in scientific notation. --- **Step 2: Recall scientific notation form** Scientific notation is: \( a \times 10^n \) where \( 1 \leq a < 10 \) and \( n \) is an integer. --- **Step 3: Place the decimal point in 16,777,216** The number is 16,777,216. To get \( a \) between 1 and 10, we move the decimal point (which is after the 6) to the left until we have a number between 1 and 10. 16,777,216 = 1.6777216 × \( 10^n \) --- **Step 4: Determine \( n \)** Original: 16,777,216. Decimal starts after the last digit: 16,777,216.0 We move the decimal point 7 places to the left to get 1.6777216. Why 7 places? Count: 1 → 6,777,216.0 (1 place) 1 6 → 7,772,16.0 (2 places) 1 6 7 → 7,721,6.0 (3 places) 1 6 7 7 → 7,216.0 (4 places) 1 6 7 7 7 → 216.0 (5 places) 1 6 7 7 7 2 → 16.0 (6 places) 1 6 7 7 7 2 1 → 6.0 (7 places) So \( n = 7 \). --- **Step 5: Write in scientific notation** 16,777,216 = 1.6777216 × 10^7 In the "e" notation used in the correct answer: 1.6777216e7 --- **Final Answer:** 1.6777216e7

4. A triangular garden is drawn on a coordinate plane with vertices at (0, 0), (12, 0), and (6, 8). If each unit on the grid represents 5 meters in real life, what is the actual area of the garden in square meters? Answer: 1200 Solution: Identify the base of the triangle on the coordinate plane. The base is between points (0, 0) and (12, 0), so the length is 12 - 0 = 12 units. Identify the height of the triangle.
   Full step-by-step solution

   Step 1: Identify the base of the triangle on the coordinate plane. The base is between points (0, 0) and (12, 0), so the length is 12 - 0 = 12 units. Step 2: Identify the height of the triangle. The height is the vertical distance from the third vertex (6, 8) to the base along y=0, which is 8 - 0 = 8 units. Step 3: Calculate the area of the triangle on the grid using the formula: Area = (1/2) × base × height = (1/2) × 12 × 8 = (1/2) × 96 = 48 square units. Step 4: Apply the scale factor. Each grid unit represents 5 meters, so for area, we square the scale factor: (5 m/unit)² = 25 m²/square unit. Step 5: Calculate the actual area: 48 square units × 25 m²/square unit = 1200 m². The actual area of the garden is 1200 square meters.

5. Mason is researching the population of a rare species of butterfly. He finds that the current population is approximately 2.7 × 10^5 butterflies. Due to conservation efforts, the population is expected to increase by a factor of 2.2 × 10^2 over the next decade. What will be the expected population of butterflies after the increase? Express your answer in proper scientific notation. Answer: 5.94 × 10^7 Solution: Write the current population and growth factor in scientific notation. Current population = 2.7 × 10^5. Growth factor = 2.2 × 10^2.
   Full step-by-step solution

   Step 1: Write the current population and growth factor in scientific notation. Current population = 2.7 × 10^5. Growth factor = 2.2 × 10^2. Step 2: Multiply the coefficients: 2.7 × 2.2 = 5.94. Step 3: Multiply the powers of 10 by adding the exponents: 10^5 × 10^2 = 10^(5+2) = 10^7. Step 4: Combine the results: 5.94 × 10^7. Step 5: Check proper scientific notation: The coefficient 5.94 is between 1 and 10, so this is correct. The expected population is 5.94 × 10^7 butterflies.

6. Write 572,000 in scientific notation. Answer: 5.72 × 10⁵ Solution: Start with 572,000. Write it as 572,000.0. Move the decimal point left until only one non-zero digit is to its left: 5.72000.
   Full step-by-step solution

   Step 1: Start with 572,000. Write it as 572,000.0. Step 2: Move the decimal point left until only one non-zero digit is to its left: 5.72000. Step 3: Count the number of places moved: 5 places. Step 4: The coefficient is 5.72 (since 1 ≤ 5.72 < 10). Step 5: The exponent is 5 (positive because the original number is large). Step 6: Write in scientific notation: 5.72 × 10⁵. The answer is 5.72 × 10⁵.

7. Hana is helping her school's science club prepare a presentation about the human body. She learns that the average human body contains approximately 3.6 × 10^4 red blood cells per microliter of blood. If an average adult has about 5.0 × 10^6 microliters of blood in their body, how many total red blood cells does an average adult have? Express your answer in proper scientific notation. Answer: 1.8 × 10^11 Solution: Write the given numbers in scientific notation. Number of red blood cells per microliter = 3.6 × 10^4 Total microliters of blood = 5.0 × 10^6 Multiply to find the total number of red blood cells.
   Full step-by-step solution

   Step 1: Write the given numbers in scientific notation. Number of red blood cells per microliter = 3.6 × 10^4 Total microliters of blood = 5.0 × 10^6 Step 2: Multiply to find the total number of red blood cells. Total = (3.6 × 10^4) × (5.0 × 10^6) Step 3: Multiply the coefficients: 3.6 × 5.0 = 18 Step 4: Multiply the powers of 10: 10^4 × 10^6 = 10^(4+6) = 10^10 Step 5: Combine: 18 × 10^10 Step 6: Check if this is in proper scientific notation. The coefficient 18 is not between 1 and 10, so we need to adjust. 18 = 1.8 × 10^1, so 18 × 10^10 = 1.8 × 10^1 × 10^10 = 1.8 × 10^(1+10) = 1.8 × 10^11 The answer is 1.8 × 10^11 red blood cells.

8. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (12, 1), (12, 8), and (2, 8). If each unit on the grid represents 3 meters, what is the actual area of the garden in square meters? Answer: 630 Solution: (2, 1), (12, 1), (12, 8), (2, 8) - (2, 1) to (12, 1) is a horizontal side - (12, 1) to (12, 8) is a vertical side - (12, 8) to (2, 8) is a horizontal side - (2, 8) to (2, 1) is a vertical side Horizontal side length = difference in x-coordinates between (2, 1) and (12, 1): Length = 12 - 2 = 10…
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Identify the coordinates and shape** The corners are: (2, 1), (12, 1), (12, 8), (2, 8) This is a rectangle because: - (2, 1) to (12, 1) is a horizontal side - (12, 1) to (12, 8) is a vertical side - (12, 8) to (2, 8) is a horizontal side - (2, 8) to (2, 1) is a vertical side --- **Step 2: Find the side lengths in grid units** Horizontal side length = difference in x-coordinates between (2, 1) and (12, 1): Length = 12 - 2 = 10 units Vertical side length = difference in y-coordinates between (12, 1) and (12, 8): Width = 8 - 1 = 7 units --- **Step 3: Calculate area in grid square units** Area in grid units = length × width = 10 × 7 = 70 square units --- **Step 4: Convert to actual meters** Each grid unit = 3 meters in real life. So: Actual length = 10 units × 3 m/unit = 30 m Actual width = 7 units × 3 m/unit = 21 m Actual area = 30 m × 21 m = 630 square meters --- **Step 5: Alternative check using scale factor for area** Since 1 unit = 3 meters, 1 square unit = (3 m) × (3 m) = 9 square meters. So actual area = 70 square units × 9 m²/square unit = 630 m². --- **Final answer:** 630