Let's break this down step by step! --- # 1. **Analyze the Rivet Shape** The rivet consists of: - **A cylindrical shaft:** Length = 10. cm, Diameter = 3. cm (radius = 1.5 cm) - **A head (modeled as a cylinder):** Height = 2. cm, Diameter = 6.4 cm (radius = 3.2 cm) --- # 2. **Calculate Volumes** ## **A. Shaft Volume (Cylinder)** \[ V_{shaft} = \pi r^2 h \] - \( r = 1.5\, \text{cm} \) - \( h = 10.\, \text{cm} \) \[ V_{shaft} = \pi (1.5)^2 \times 10. = \pi \times 2.25 \times 10. = 22.5\pi \approx 70.7\, \text{cm}^3 \] --- ## **B. Head Volume (Cylinder)** \[ V_{head} = \pi r^2 h \] - \( r = 3.2\, \text{cm} \) - \( h = 2.\, \text{cm} \) \[ V_{head} = \pi (3.2)^2 \times 2. = \pi \times 10.24 \times 2. = 20.48\pi \approx 64.3\, \text{cm}^3 \] --- ## **C. Subtract Shaft Section in Head** The head sits on top of the shaft, so the shaft's cylinder "intrudes" into the head. Subtract the volume of the shaft inside the head (cylinder of radius \(1.5\,\text{cm}\), height \(2.\,\text{cm}\)): \[ V_{overlap} = \pi (1.5)^2 \times 2. = \pi \times 2.25 \times 2. = 4.5\pi \approx 14.1\, \text{cm}^3 \] --- ## **D. Total Volume** \[ V_{total} = V_{shaft} + V_{head} - V_{overlap} \] \[ V_{total} \approx 70.7 + 64.3 - 14.1 = 120.9\, \text{cm}^3 \] --- # 3. **Find Weight in Pounds** Given: **Steel density = .0173 lb/cm³** \[ \text{Weight (lb)} = V_{total} \times \text{density} \] \[ \text{Weight (lb)} = 120.9 \times .0173 \approx 2.09\, \text{lb} \] --- # 4. **Convert Pounds to Ounces** There are **16 ounces in 1 pound**: \[ \text{Weight (oz)} = 2.09 \times 16 \approx 33.4\, \text{oz} \] --- # **Final Answer** > **The weight of the steel rivet is approximately \( \boxed{33.4\, \text{oz}} \) (ounces).** --- If you have any questions about the steps or want to see the math for a different head shape, just ask!