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
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.
To calculate the volume of the rectangular box, you need to multiply the lengths of its three sides together.
Given the lengths:
Length = 1.80 + 0.01 cm
Width = 2.05 + 0.03 cm
Height = 3.10 + 0.01 cm
You can calculate the volume as follows:
Volume = Length x Width x Height
Volume = (1.80 + 0.01) cm x (2.05 + 0.03) cm x (3.10 + 0.01) cm
Volume = 1.81 cm x 2.08 cm x 3.11 cm
Volume = 11.90948 cm³ (rounded to five decimal places)
The approximate uncertainty in the volume can be calculated by adding the absolute uncertainties of the three dimensions.
Uncertainty = 0.01 cm + 0.03 cm + 0.01 cm
Uncertainty = 0.05 cm
Therefore, the approximate uncertainty in the volume of the box is 0.05 cm³.
To calculate the volume of the rectangular box, you need to multiply the lengths of its three sides. In this case, the lengths of the sides are given as:
Length of side 1: 1.80 + 0.01 cm
Length of side 2: 2.05 + 0.03 cm
Length of side 3: 3.10 + 0.01 cm
To find the actual lengths of the sides, you can subtract the uncertainty values from the given measurements:
Length of side 1 = 1.80 - 0.01 cm = 1.79 cm
Length of side 2 = 2.05 - 0.03 cm = 2.02 cm
Length of side 3 = 3.10 - 0.01 cm = 3.09 cm
Now you can calculate the volume by multiplying the lengths of the sides:
Volume = Length side 1 × Length side 2 × Length side 3
Volume = 1.79 cm × 2.02 cm × 3.09 cm
Volume ≈ 11.4482 cm³ (rounded to four decimal places)
To find the approximate uncertainty in the volume, you need to consider the uncertainties in the measurements. Since the uncertainties are given as ±0.01 cm and ±0.03 cm, respectively, you'll use the maximum difference from the actual measurements:
Maximum uncertainty = Max(±0.01 cm, ±0.03 cm) = ±0.03 cm
To calculate the approximate uncertainty in the volume, you'll use the formula for propagating uncertainties through multiplication:
Uncertainty in volume = Volume × (Uncertainty in length side 1 / Length side 1 + Uncertainty in length side 2 / Length side 2 + Uncertainty in length side 3 / Length side 3)
Uncertainty in volume = 11.4482 cm³ × (0.03 cm / 1.79 cm + 0.03 cm / 2.02 cm + 0.03 cm / 3.09 cm)
Uncertainty in volume ≈ 0.487 cm³ (rounded to three decimal places)
Therefore, the volume of the rectangular box is approximately 11.4482 cm³, and the approximate uncertainty in the volume is 0.487 cm³.