If the length of a side of a cube is measured as 2.5 cm with an uncertainty of 0.1 cm, what is the volume of the cube with its uncertainty?

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Q: If the length of a side of a cube is measured as 2.5 cm with an uncertainty of 0.1 cm, what is the volume of the cube with its uncertainty?

Solution: Volume = (2.5 cm)³ = 15.625 cm³. The uncertainty in volume can be calculated using the formula for propagation of uncertainty.

Steps: 10

Step 1: Understand that the volume of a cube is calculated using the formula Volume = side length³.
Step 2: Identify the side length of the cube, which is given as 2.5 cm.
Step 3: Calculate the volume of the cube by cubing the side length: Volume = (2.5 cm)³.
Step 4: Perform the calculation: 2.5 × 2.5 = 6.25, and then 6.25 × 2.5 = 15.625 cm³.
Step 5: Now, we need to calculate the uncertainty in the volume. The formula for uncertainty in volume is: ΔV = 3 × (side length)² × ΔL, where ΔL is the uncertainty in the side length.
Step 6: Identify the uncertainty in the side length, which is given as 0.1 cm.
Step 7: Calculate the uncertainty in volume: ΔV = 3 × (2.5 cm)² × 0.1 cm.
Step 8: First, calculate (2.5 cm)² = 6.25 cm².
Step 9: Now, multiply: ΔV = 3 × 6.25 cm² × 0.1 cm = 1.875 cm³.
Step 10: Combine the volume and its uncertainty: The volume of the cube is 15.625 cm³ with an uncertainty of ±1.875 cm³.