Let’s solve each problem step by step, following your requested structure and definition of parameters. --- ## **Question 1** ### **Given:** - \( a = 5 \pm 1 \) cm - \( b = 18 \pm 2 \) cm - \( c = 12 \pm 1 \) cm - \( t = 3.0 \pm 0.5 \) s - \( m = 18 \pm 1 \) gr - Formula to use: \( d = \frac{ma}{t} \) --- ### **Step 1: Calculate the Value of \( d \)** \[ d = \frac{ma}{t} = \frac{18 \times 5}{3.0} = \frac{90}{3.0} = 30 \] **Explanation:** Multiply the values of \( m \) and \( a \), and then divide by \( t \) to get the central value for \( d \). --- ### **Step 2: Calculate the Uncertainty in \( d \)** Relative uncertainty formula: \[ \frac{\Delta d}{d} = \frac{\Delta m}{m} + \frac{\Delta a}{a} + \frac{\Delta t}{t} \] Calculate each relative uncertainty: - \( \frac{\Delta m}{m} = \frac{1}{18} \approx 0.056 \) - \( \frac{\Delta a}{a} = \frac{1}{5} = 0.20 \) - \( \frac{\Delta t}{t} = \frac{0.5}{3.0} \approx 0.167 \) Sum them: \[ \frac{\Delta d}{d} = 0.056 + 0.20 + 0.167 = 0.423 \] \[ \Delta d = d \times 0.423 = 30 \times 0.423 \approx 12.7 \] Rounded to 2 significant figures: \( \Delta d \approx 13 \). **Explanation:** Add the relative uncertainties for multiplication/division, then multiply by the calculated value to get the absolute uncertainty. --- ### **Final Answer Summary:** \[ \boxed{d = 30 \pm 13} \] This result includes the calculated value and its uncertainty, following the rules of uncertainty propagation. --- ## **Question 2** ### **Given:** - \( D = 4.0 \pm 0.2 \) cm - Formula: \( V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{D}{2}\right)^3 \) --- ### **Step 1: Calculate the Volume** \[ V = \frac{4}{3} \pi \left(\frac{4.0}{2}\right)^3 = \frac{4}{3} \pi (2.0)^3 = \frac{4}{3} \pi \times 8.0 = \frac{32}{3} \pi \approx 33.51 \text{ cm}^3 \] **Explanation:** First, calculate the radius by dividing the diameter by 2, then cube the radius, multiply by \(\frac{4}{3}\pi\). --- ### **Step 2: Calculate the Uncertainty in Volume** Relative uncertainty for a power: \[ \frac{\Delta V}{V} = 3 \times \frac{\Delta D}{D} \] \[ \frac{\Delta D}{D} = \frac{0.2}{4.0} = 0.05 \] \[ \frac{\Delta V}{V} = 3 \times 0.05 = 0.15 \] \[ \Delta V = V \times 0.15 = 33.51 \times 0.15 \approx 5.0 \] **Explanation:** Since the diameter is cubed, multiply the relative uncertainty by 3. Use this to calculate the absolute uncertainty in the volume. --- ### **Final Answer Summary:** \[ \boxed{V_{\text{sphere}} = 33.5 \pm 5.0 \text{ cm}^3} \] Volume and its uncertainty are reported based on the propagation of uncertainties for powers. --- **Parameter Review:** - **Relevancy:** Each step directly addresses the computation and propagation of uncertainty. - **Completeness:** All calculations, steps, and explanations are included. - **Accuracy:** Mathematical procedures follow accepted rules for uncertainty propagation. - **Clarity:** Steps and explanations use clear, concise language. - **Structure:** Logical step-by-step format aids understanding. - **Voice:** Human-like, instructional tone is maintained throughout.

**Question 1** ### **Given:** - \( a = 5 \pm 1\, \text{cm} \) - \( b = 18 \pm 2\, \text{cm} \) - \( c = 12 \pm 1\, \text{cm} \) - \( t = 3.0 \pm 0.5\, \text{s} \) - \( m = 18 \pm 1\, \text{g} \) - Formula: \( d = \frac{m a}{t} \) --- ### **Step 1: Calculate the central value of \( d \)** \[ d = \frac{m a}{t} = \frac{18\, \text{g} \times 5\, \text{cm}}{3.0\, \text{s}} = \frac{90\, \text{g} \cdot \text{cm}}{3.0\, \text{s}} = 30\, \text{g} \cdot \text{cm/s} \] **Explanation:** Multiply mass \( m \) and length \( a \), then divide by time \( t \) to find the value of \( d \) with units of grams times centimeters per second. --- ### **Step 2: Calculate the uncertainty in \( d \)** Relative uncertainties: \[ \frac{\Delta m}{m} = \frac{1}{18} \approx 0.056 \] \[ \frac{\Delta a}{a} = \frac{1}{5} = 0.20 \] \[ \frac{\Delta t}{t} = \frac{0.5}{3.0} \approx 0.167 \] Total relative uncertainty: \[ \frac{\Delta d}{d} = 0.056 + 0.20 + 0.167 = 0.423 \] Absolute uncertainty: \[ \Delta d = d \times 0.423 = 30 \times 0.423 \approx 12.7 \] Rounding to two significant figures: \[ \Delta d \approx 13\, \text{g} \cdot \text{cm/s} \] **Explanation:** Sum the relative uncertainties for multiplication/division, then multiply by the calculated \( d \) to find the absolute uncertainty, including units. --- ### **Final answer:** \[ \boxed{ d = 30 \pm 13\, \text{g} \cdot \text{cm/s} } \] --- **Question 2** ### **Given:** - Diameter \( D = 4.0 \pm 0.2\, \text{cm} \) ### **Step 1: Calculate the volume of the sphere** Radius: \[ r = \frac{D}{2} = \frac{4.0\, \text{cm}}{2} = 2.0\, \text{cm} \] Volume: \[ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2.0\, \text{cm})^3 = \frac{4}{3} \pi \times 8.0\, \text{cm}^3 \approx 33.51\, \text{cm}^3 \] **Explanation:** Calculate the radius from the diameter, then apply the sphere volume formula to obtain the volume with units of cubic centimeters. --- ### **Step 2: Calculate the uncertainty in volume** Since volume depends on the cube of the diameter, the relative uncertainty: \[ \frac{\Delta V}{V} = 3 \times \frac{\Delta D}{D} = 3 \times \frac{0.2\, \text{cm}}{4.0\, \text{cm}} = 3 \times 0.05 = 0.15 \] Absolute uncertainty: \[ \Delta V = V \times 0.15 \approx 33.51\, \text{cm}^3 \times 0.15 \approx 5.0\, \text{cm}^3 \] **Explanation:** Because volume scales with the cube of diameter, multiply the relative uncertainty by 3, then compute the absolute uncertainty accordingly. --- ### **Final answer:** \[ \boxed{ V = 33.5 \pm 5.0\, \text{cm}^3 } \] --- **Summary:** - The calculated \( d \) is **30 ± 13 g·cm/s**. - The sphere volume is **33.5 ± 5.0 cm³**. All uncertainties are propagated following standard rules, and units are explicitly included for clarity.