1. **Problem Statement:** Calculate the volume and surface area of a composite shape consisting of a rectangular prism and a cone attached to its side. 2. **Given Dimensions:** - Rectangular prism: height $H=2.0$ m, width $W=1.0$ m, depth $D=1.0$ m - Cone: height $h=1.5$ m, base radius $r=0.5$ m 3. **Volume Calculation:** The total volume $V$ is the sum of the prism volume $V_{prism}$ and the cone volume $V_{cone}$. - Prism volume formula: $$V_{prism} = L \times W \times H$$ - Cone volume formula: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ Calculate each: $$V_{prism} = 1.0 \times 1.0 \times 2.0 = 2.0$$ $$V_{cone} = \frac{1}{3} \pi (0.5)^2 (1.5) = \frac{1}{3} \pi \times 0.25 \times 1.5 = \frac{1}{3} \pi \times 0.375 = 0.125\pi$$ Approximate $V_{cone}$: $$0.125 \times 3.1416 \approx 0.3927$$ Total volume: $$V = V_{prism} + V_{cone} = 2.0 + 0.3927 = 2.3927 \text{ m}^3$$ 4. **Surface Area Calculation:** The total surface area $SA$ is the sum of the prism surface area minus the area where the cone attaches plus the cone's lateral surface area and base area. - Prism surface area formula: $$SA_{prism} = 2(LW + LH + WH)$$ Calculate: $$SA_{prism} = 2(1.0 \times 1.0 + 1.0 \times 2.0 + 1.0 \times 2.0) = 2(1 + 2 + 2) = 2 \times 5 = 10$$ - Area of prism face where cone attaches (width $\times$ height): $$A_{attach} = W \times H = 1.0 \times 2.0 = 2.0$$ - Cone lateral surface area formula: $$SA_{cone, lateral} = \pi r s$$ where slant height $s = \sqrt{r^2 + h^2} = \sqrt{0.5^2 + 1.5^2} = \sqrt{0.25 + 2.25} = \sqrt{2.5} \approx 1.5811$$ Calculate lateral area: $$SA_{cone, lateral} = \pi \times 0.5 \times 1.5811 = 0.5\pi \times 1.5811 = 0.79055\pi \approx 2.483$$ - Cone base area: $$SA_{cone, base} = \pi r^2 = \pi \times 0.5^2 = \pi \times 0.25 = 0.25\pi \approx 0.7854$$ 5. **Total Surface Area:** $$SA = SA_{prism} - A_{attach} + SA_{cone, lateral} + SA_{cone, base}$$ $$SA = 10 - 2 + 2.483 + 0.7854 = 8 + 3.2684 = 11.2684 \text{ m}^2$$ **Final answers:** - Volume $V \approx 2.39$ m$^3$ - Surface Area $SA \approx 11.27$ m$^2$