Question
The sides of a small resctangular box are measures to be 1.80+0.01 cm long, 2.05+0.03 cm long and 3.10+0.01 cm long. calculate the boxes volume and approximate uncertainty in cubic centimeters
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GPT 3.5
To calculate the volume of the rectangular box, we multiply the lengths of its sides:
Volume = (1.80+0.01) cm * (2.05+0.03) cm * (3.10+0.01) cm
Volume = (1.81 cm) * (2.08 cm) * (3.11 cm)
Volume ≈ 11.73448 cm³
The approximate uncertainty in the volume can be calculated using the formula for propagation of uncertainty:
Uncertainty in Volume = (dV/dl₁) * Δl₁ + (dV/dl₂) * Δl₂ + (dV/dl₃) * Δl₃
Where:
dV/dl₁, dV/dl₂, and dV/dl₃ represent the partial derivatives of the volume with respect to each side length, and Δl₁, Δl₂, and Δl₃ represent the uncertainties in each side length. Since all the side lengths have a small uncertainty of ± 0.01 cm, we can assume Δl₁ = Δl₂ = Δl₃ = 0.01 cm.
Partial derivatives:
(dV/dl₁) = (2.08 cm) * (3.11 cm) ≈ 6.4568 cm²
(dV/dl₂) = (1.81 cm) * (3.11 cm) ≈ 5.6251 cm²
(dV/dl₃) = (1.81 cm) * (2.08 cm) ≈ 3.7608 cm²
Uncertainty in Volume ≈ (6.4568 cm²) * (0.01 cm) + (5.6251 cm²) * (0.01 cm) + (3.7608 cm²) * (0.01cm)
Uncertainty in Volume ≈ 0.6457 cm³ + 0.0563 cm³ + 0.0376 cm³
Uncertainty in Volume ≈ 0.7396 cm³
Therefore, the box's volume is approximately 11.73448 cm³ and the approximate uncertainty in the volume is 0.7396 cm³.
Volume = (1.80+0.01) cm * (2.05+0.03) cm * (3.10+0.01) cm
Volume = (1.81 cm) * (2.08 cm) * (3.11 cm)
Volume ≈ 11.73448 cm³
The approximate uncertainty in the volume can be calculated using the formula for propagation of uncertainty:
Uncertainty in Volume = (dV/dl₁) * Δl₁ + (dV/dl₂) * Δl₂ + (dV/dl₃) * Δl₃
Where:
dV/dl₁, dV/dl₂, and dV/dl₃ represent the partial derivatives of the volume with respect to each side length, and Δl₁, Δl₂, and Δl₃ represent the uncertainties in each side length. Since all the side lengths have a small uncertainty of ± 0.01 cm, we can assume Δl₁ = Δl₂ = Δl₃ = 0.01 cm.
Partial derivatives:
(dV/dl₁) = (2.08 cm) * (3.11 cm) ≈ 6.4568 cm²
(dV/dl₂) = (1.81 cm) * (3.11 cm) ≈ 5.6251 cm²
(dV/dl₃) = (1.81 cm) * (2.08 cm) ≈ 3.7608 cm²
Uncertainty in Volume ≈ (6.4568 cm²) * (0.01 cm) + (5.6251 cm²) * (0.01 cm) + (3.7608 cm²) * (0.01cm)
Uncertainty in Volume ≈ 0.6457 cm³ + 0.0563 cm³ + 0.0376 cm³
Uncertainty in Volume ≈ 0.7396 cm³
Therefore, the box's volume is approximately 11.73448 cm³ and the approximate uncertainty in the volume is 0.7396 cm³.
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