Scientific Operations

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Scientific Operations · Grade 7 · Worksheet 3

1. A triangular prism is drawn with a triangular base that is a right triangle. The legs of the triangular base are 6 cm and 8 cm, and the height of the prism is 15 cm. What is the total surface area of the prism in square centimeters? Answer: 408 Solution: Find the area of one triangular base. The triangular base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 × 8 = 24 cm² Find the hypotenuse of the triangular base.
   Full step-by-step solution

   Step 1: Find the area of one triangular base. The triangular base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 × 8 = 24 cm² Step 2: Find the hypotenuse of the triangular base. Using Pythagorean theorem: hypotenuse² = 6² + 8² = 36 + 64 = 100 hypotenuse = √100 = 10 cm Step 3: Find the area of the three rectangular lateral faces. - Face 1: 6 cm × 15 cm = 90 cm² - Face 2: 8 cm × 15 cm = 120 cm² - Face 3: 10 cm × 15 cm = 150 cm² Step 4: Calculate total lateral surface area. Total lateral area = 90 + 120 + 150 = 360 cm² Step 5: Calculate total surface area. Total surface area = lateral area + 2 × base area Total surface area = 360 + 2 × 24 = 360 + 48 = 408 cm² The answer is 408.

2. (7.5 × 10⁷) ÷ (1.5 × 10³) = ? Answer: 50000 Solution: Write the expression: (7.5 × 10⁷) ÷ (1.5 × 10³) Divide the coefficients: 7.5 ÷ 1.5 = 5 Divide the powers of 10: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Combine the results: 5 × 10⁴ Convert to standard form: 5 × 10,000 = 50,000 The answer is 50000.
   Full step-by-step solution

   Step 1: Write the expression: (7.5 × 10⁷) ÷ (1.5 × 10³) Step 2: Divide the coefficients: 7.5 ÷ 1.5 = 5 Step 3: Divide the powers of 10: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Step 4: Combine the results: 5 × 10⁴ Step 5: Convert to standard form: 5 × 10,000 = 50,000 The answer is 50000.

3. (6.8 × 10⁷) ÷ (2.0 × 10³) = ? Answer: 34000 Solution: Write the expression: (6.8 × 10⁷) ÷ (2.0 × 10³) Divide the coefficients: 6.8 ÷ 2.0 = 3.4 Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Combine: 3.4 × 10⁴ Convert to standard form: 3.4 × 10,000 = 34,000 The answer is 34000.
   Full step-by-step solution

   Step 1: Write the expression: (6.8 × 10⁷) ÷ (2.0 × 10³) Step 2: Divide the coefficients: 6.8 ÷ 2.0 = 3.4 Step 3: Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴ Step 4: Combine: 3.4 × 10⁴ Step 5: Convert to standard form: 3.4 × 10,000 = 34,000 The answer is 34000.

4. 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

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). What is the length of the hypotenuse of this triangle? Answer: 10 Solution: Identify the side lengths from the coordinates. The points (0,0) and (6,0) have the same y-coordinate, so the horizontal leg length is |6 - 0| = 6 units.
   Full step-by-step solution

   Step 1: Identify the side lengths from the coordinates. The points (0,0) and (6,0) have the same y-coordinate, so the horizontal leg length is |6 - 0| = 6 units. The points (0,0) and (0,8) have the same x-coordinate, so the vertical leg length is |8 - 0| = 8 units. Step 2: Apply the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. Step 3: Substitute the values: 6² + 8² = c² Step 4: Calculate: 36 + 64 = 100 Step 5: Find c by taking the square root: c = sqrt(100) = 10 Step 6: The hypotenuse is 10 units long.

6. A triangular prism is drawn with a right triangular base. The triangular base has legs measuring 6 cm and 8 cm, and the prism has a height (length) of 15 cm. What is the total surface area of this prism in square centimeters? Answer: 408 Solution: Find the hypotenuse of the triangular base using the Pythagorean theorem. The legs are 6 cm and 8 cm, so: Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm Calculate the area of the two triangular bases.
   Full step-by-step solution

   Step 1: Find the hypotenuse of the triangular base using the Pythagorean theorem. The legs are 6 cm and 8 cm, so: Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm Step 2: Calculate the area of the two triangular bases. Area of one triangle = (1/2) × base × height = (1/2) × 6 × 8 = 24 cm² Area of two triangles = 2 × 24 = 48 cm² Step 3: Calculate the area of the three rectangular faces. Face 1: 6 cm × 15 cm = 90 cm² Face 2: 8 cm × 15 cm = 120 cm² Face 3: 10 cm × 15 cm = 150 cm² Step 4: Add all areas together. Total surface area = 48 + 90 + 120 + 150 = 408 cm² The answer is 408.

7. A research scientist is studying bacterial growth in a lab. She observes that a colony of bacteria doubles in size every 3 hours. If the colony starts with 5.2 × 10³ bacteria, how many bacteria will there be after 12 hours? Express your answer in scientific notation. Answer: 8.32 × 10⁴ Solution: The colony starts with \( 5.2 \times 10^3 \) bacteria. Doubling time = 3 hours. We want the number after 12 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The colony starts with \( 5.2 \times 10^3 \) bacteria. Doubling time = 3 hours. We want the number after 12 hours. --- **Step 2: Determine the number of doubling periods** Each doubling period = 3 hours. Total time = 12 hours. Number of doublings \( n \) = \( \frac{12}{3} = 4 \). So the colony doubles 4 times. --- **Step 3: Apply the doubling formula** Initial amount: \( N_0 = 5.2 \times 10^3 \) After \( n \) doublings: \( N = N_0 \times 2^n \) Here: \( N = (5.2 \times 10^3) \times 2^4 \) --- **Step 4: Calculate \( 2^4 \)** \( 2^4 = 16 \) So: \( N = (5.2 \times 10^3) \times 16 \) --- **Step 5: Multiply \( 5.2 \) by \( 16 \)** \( 5.2 \times 16 = 83.2 \) So: \( N = 83.2 \times 10^3 \) --- **Step 6: Convert to proper scientific notation** \( 83.2 \times 10^3 = 8.32 \times 10^1 \times 10^3 \) \( = 8.32 \times 10^{4} \) --- **Step 7: Final answer** After 12 hours, the number of bacteria is \( 8.32 \times 10^4 \).