Scientific Notation

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Scientific Notation · Grade 7 · Worksheet 2

1. Lily is studying the solar system and learns that the distance from Earth to Mars is approximately 7.83 × 10^7 kilometers. If a spacecraft travels at a constant speed of 2.61 × 10^4 kilometers per hour, how many hours would it take to reach Mars? Express your answer in scientific notation. Answer: 3.00 × 10^3 Solution: Write the distance and speed: Distance = 7.83 × 10^7 km, Speed = 2.61 × 10^4 km/h Use the formula: Time = Distance ÷ Speed Time = (7.83 × 10^7) ÷ (2.61 × 10^4) Divide the coefficients: 7.83 ÷ 2.61 = 3.00 Subtract the exponents: 10^(7-4) = 10^3 Combine the results: 3.00 × 10^3 hours The answer is…
   Full step-by-step solution

   Step 1: Write the distance and speed: Distance = 7.83 × 10^7 km, Speed = 2.61 × 10^4 km/h Step 2: Use the formula: Time = Distance ÷ Speed Step 3: Time = (7.83 × 10^7) ÷ (2.61 × 10^4) Step 4: Divide the coefficients: 7.83 ÷ 2.61 = 3.00 Step 5: Subtract the exponents: 10^(7-4) = 10^3 Step 6: Combine the results: 3.00 × 10^3 hours The answer is 3.00 × 10^3 hours.

2. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If each unit on the grid represents 3 meters in real life, what is the actual area of the garden in square meters? Answer: 216 Solution: Identify the coordinates of the garden's vertices. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). Determine the length and width in coordinate units.
   Full step-by-step solution

   Step 1: Identify the coordinates of the garden's vertices. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). Step 2: Determine the length and width in coordinate units. The points (2, 1) and (8, 1) have the same y-coordinate, so the horizontal distance is: 8 - 2 = 6 units. The points (2, 1) and (2, 5) have the same x-coordinate, so the vertical distance is: 5 - 1 = 4 units. So the rectangle is 6 units long and 4 units wide on the grid. Step 3: Calculate the area in grid square units. Area in grid units = length × width = 6 × 4 = 24 square units. Step 4: Convert to real-life measurements. Each grid unit represents 3 meters in real life. So 1 grid unit length = 3 meters. 1 grid square unit = (3 meters) × (3 meters) = 9 square meters. Step 5: Calculate the actual area in square meters. Actual area = (Area in grid units) × (Square meters per grid square unit) Actual area = 24 × 9 = 216 square meters. Final answer: 216

3. Noah is learning about the populations of different countries. He finds that the population of Japan is approximately 1.26 × 10^8 people, and the population of Mexico is approximately 1.28 × 10^8 people. What is the combined population of Japan and Mexico? Express your answer in proper scientific notation. Answer: 2.54 × 10^8 Solution: Write the populations in scientific notation. Japan: 1.26 × 10^8 Mexico: 1.28 × 10^8 Since both numbers have the same power of 10 (10^8), add the coefficients. 1.26 + 1.28 = 2.54 Keep the power of 10 the same.
   Full step-by-step solution

   Step 1: Write the populations in scientific notation. Japan: 1.26 × 10^8 Mexico: 1.28 × 10^8 Step 2: Since both numbers have the same power of 10 (10^8), add the coefficients. 1.26 + 1.28 = 2.54 Step 3: Keep the power of 10 the same. 2.54 × 10^8 Step 4: Check if the coefficient is in proper form (1 ≤ coefficient < 10). 2.54 is between 1 and 10, so it is already in proper scientific notation. The combined population of Japan and Mexico is 2.54 × 10^8 people.

4. A research team is studying the growth of algae in a lake. They measure that the algae covers 2.5 × 10^4 square meters initially and grows at a rate of 8% per day. How many square meters will the algae cover after 7 days? Express your answer in proper scientific notation. Answer: 4.3 × 10^4 Solution: Identify the initial area: 2.5 × 10^4 = 25,000 square meters Identify the growth rate: 8% per day, which means multiplying by 1.08 each day Calculate the growth over 7 days: 25,000 × (1.08)^7 Calculate 1.08^7: 1.08 × 1.08 = 1.1664, 1.1664 × 1.08 = 1.259712, 1.259712 × 1.08 = 1.360489, 1.360489 ×…
   Full step-by-step solution

   Step 1: Identify the initial area: 2.5 × 10^4 = 25,000 square meters Step 2: Identify the growth rate: 8% per day, which means multiplying by 1.08 each day Step 3: Calculate the growth over 7 days: 25,000 × (1.08)^7 Step 4: Calculate 1.08^7: 1.08 × 1.08 = 1.1664, 1.1664 × 1.08 = 1.259712, 1.259712 × 1.08 = 1.360489, 1.360489 × 1.08 = 1.469328, 1.469328 × 1.08 = 1.586874, 1.586874 × 1.08 = 1.713824 Step 5: Multiply: 25,000 × 1.713824 = 42,845.6 Step 6: Convert to scientific notation: 42,845.6 = 4.28456 × 10^4 Step 7: Round to 2 significant figures: 4.3 × 10^4 Therefore, the algae will cover 4.3 × 10^4 square meters after 7 days.

5. (4.5 × 10⁵) ÷ (9 × 10²) = ? Answer: 500 Solution: Step 1: Write the division of numbers in scientific notation: (4.5 × 10⁵) ÷ (9 × 10²) Step 2: Divide the coefficients: 4.5 ÷ 9 = 0.5 Step 3: Divide the powers of 10: 10⁵ ÷ 10² = 10³ (subtract exponents: 5 - 2 = 3) Step 4: Multiply the results: 0.5 × 10³ = 500 Step 5: Verify: 500 × (9 × 10²) =…
   Full step-by-step solution

   Step 1: Write the division of numbers in scientific notation: (4.5 × 10⁵) ÷ (9 × 10²) Step 2: Divide the coefficients: 4.5 ÷ 9 = 0.5 Step 3: Divide the powers of 10: 10⁵ ÷ 10² = 10³ (subtract exponents: 5 - 2 = 3) Step 4: Multiply the results: 0.5 × 10³ = 500 Step 5: Verify: 500 × (9 × 10²) = 500 × 900 = 450,000 = 4.5 × 10⁵ The answer is 500.

6. Tane is researching the mass of different objects in the solar system. He finds that the mass of Earth's moon is approximately 7.35 × 10^22 kilograms. The mass of the dwarf planet Pluto is about 1.31 × 10^22 kilograms. What is the total mass of the moon and Pluto combined? Express your answer in proper scientific notation. Answer: 8.66 × 10^22 Solution: Write the masses in scientific notation. Mass of moon = 7.35 × 10^22 kg Mass of Pluto = 1.31 × 10^22 kg Since both numbers have the same exponent (10^22), add the coefficients.
   Full step-by-step solution

   Step 1: Write the masses in scientific notation. Mass of moon = 7.35 × 10^22 kg Mass of Pluto = 1.31 × 10^22 kg Step 2: Since both numbers have the same exponent (10^22), add the coefficients. 7.35 + 1.31 = 8.66 Step 3: Keep the same power of ten. 8.66 × 10^22 Step 4: Check that the answer is in proper scientific notation. The coefficient 8.66 is between 1 and 10, so it is correct. The total mass is 8.66 × 10^22 kilograms.

7. A rectangular prism is drawn with dimensions: length = 12 cm, width = 8 cm, and height = 15 cm. If you were to draw a net of this prism, what would be the total surface area of all the faces in the net? Answer: 792 cm² Solution: Identify the formula for surface area of a rectangular prism: SA = 2(lw + lh + wh) Substitute the given dimensions: l = 12 cm, w = 8 cm, h = 15 cm Calculate lw = 12 × 8 = 96 Calculate lh = 12 × 15 = 180 Calculate wh = 8 × 15 = 120 Add the three products: 96 + 180 + 120 = 396 Multiply by 2: 2 ×…
   Full step-by-step solution

   Step 1: Identify the formula for surface area of a rectangular prism: SA = 2(lw + lh + wh) Step 2: Substitute the given dimensions: l = 12 cm, w = 8 cm, h = 15 cm Step 3: Calculate lw = 12 × 8 = 96 Step 4: Calculate lh = 12 × 15 = 180 Step 5: Calculate wh = 8 × 15 = 120 Step 6: Add the three products: 96 + 180 + 120 = 396 Step 7: Multiply by 2: 2 × 396 = 792 Step 8: Include units: 792 cm² The answer is 792 cm².

8. A triangular prism is drawn with a right triangle base where the legs measure 6 cm and 8 cm, and the prism height is 15 cm. What is the volume of the prism in cubic centimeters? Answer: 360 Solution: Find the area of the triangular base. The base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 cm × 8 cm = (1/2) × 48 cm² = 24 cm² Calculate the volume of the prism.
   Full step-by-step solution

   Step 1: Find the area of the triangular base. The base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 cm × 8 cm = (1/2) × 48 cm² = 24 cm² Step 2: Calculate the volume of the prism. Volume of prism = area of base × height of prism = 24 cm² × 15 cm = 360 cm³ The answer is 360.