Question

for questions 1 and 2, find the surface area of each composite figure. round to the nearest tenth if necessary.

1. 4.2 m 1.9 m 3.5 m
2. 10 ft 10 ft 6 ft 11 ft 16 ft 17 ft

for questions 3 - 6, find the volume of the composite figures. round to the nearest tenth if necessary.

1. 23 mm 25 mm 12 mm 19 mm
2. 6.8 m 1 m 1 m 7.3 m 4.6 m
3. 10 cm 6 cm
4. 1.5 in 5.8 in 6 in

Answer:

1. Question 1: Surface - area of cone - cylinder composite figure

• Explanation:
• Step 1: Find the radius of the base
• The diameter of the base of the cone and cylinder is \(d = 3.5m\), so the radius \(r=\frac{3.5}{2}=1.75m\).
• Step 2: Calculate the lateral - area of the cone
• The formula for the lateral - area of a cone is \(L_{cone}=\pi rl\), where \(l = 4.2m\) and \(r = 1.75m\). So \(L_{cone}=\pi\times1.75\times4.2\approx23.1m^{2}\).
• Step 3: Calculate the lateral - area of the cylinder
• The formula for the lateral - area of a cylinder is \(L_{cylinder}=2\pi rh\), where \(h = 1.9m\) and \(r = 1.75m\). So \(L_{cylinder}=2\pi\times1.75\times1.9\approx20.9m^{2}\).
• Step 4: Calculate the area of the base of the cylinder
• The formula for the area of a circle is \(A=\pi r^{2}\), with \(r = 1.75m\). So \(A=\pi\times(1.75)^{2}\approx9.6m^{2}\).
• Step 5: Calculate the total surface area
• The total surface area \(S = L_{cone}+L_{cylinder}+A\).
• \(S\approx23.1 + 20.9+9.6=53.6m^{2}\).
• Answer: \(53.6m^{2}\)

1. Question 2: Surface - area of a composite figure (rectangular prism and triangular prism)

• Explanation:
• Step 1: Analyze the rectangular - prism part
• The rectangular prism has dimensions \(l = 16ft\), \(w = 17ft\), \(h = 11ft\).
• The surface area of a rectangular prism \(S_{1}=2(lw+lh + wh)\).
• \(S_{1}=2(16\times17 + 16\times11+17\times11)=2(272+176 + 187)=2\times635 = 1270ft^{2}\).
• Step 2: Analyze the triangular - prism part
• The triangular base has base \(b = 10ft\) and height \(h_{1}=6ft\). The length of the triangular prism along the length of the composite figure is \(l = 17ft\).
• The area of the two triangular bases \(A_{triangles}=2\times\frac{1}{2}\times10\times6 = 60ft^{2}\).
• The lateral - area of the triangular prism: The three rectangular faces have areas \(10\times17\), \( \sqrt{10^{2}-6^{2}}\times17\) (slant - height of the triangular face is \(\sqrt{10^{2}-6^{2}} = 8ft\)), and \(10\times17\).
• The lateral - area \(L_{triangular}=17\times(10 + 8+10)=476ft^{2}\).
• Step 3: Calculate the total surface area
• The total surface area \(S=S_{1}+A_{triangles}+L_{triangular}\).
• \(S = 1270+60 + 476=1806ft^{2}\).
• Answer: \(1806ft^{2}\)

1. Question 3: Volume of a composite figure (rectangular prism and triangular prism)

• Explanation:
• Step 1: Calculate the volume of the rectangular - prism part
• The rectangular prism has dimensions \(l = 25mm\), \(w = 19mm\), \(h = 23mm\). The volume of a rectangular prism \(V_{1}=lwh=25\times19\times23=10825mm^{3}\).
• Step 2: Calculate the volume of the triangular - prism part
• The triangular base has base \(b = 12mm\) and height \(h_{1}=19mm\), and length \(l = 25mm\). The volume of a triangular prism \(V_{2}=\frac{1}{2}bh_{1}l=\frac{1}{2}\times12\times19\times25 = 2850mm^{3}\).
• Step 3: Calculate the total volume
• The total volume \(V = V_{1}+V_{2}=10825+2850 = 13675mm^{3}\).
• Answer: \(13675mm^{3}\)

1. Question 4: Volume of a composite figure (cone and hemisphere)

• Explanation:
• Step 1: Calculate the volume of the cone
• The cone has radius \(r = 6cm\) and height \(h = 10cm\). The volume of a cone \(V_{1}=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times6^{2}\times10 = 120\pi cm^{3}\).
• Step 2: Calculate the volume of the hemisphere
• The volume of a hemisphere \(V_{2}=\frac{2}{3}\pi r^{3}\), with \(r = 6cm\). So \(V_{2}=\frac{2}{3}\pi\times6^{3}=144\pi cm^{3}\).
• Step 3: Calculate the total volume
• The total volume \(V=V_{1}+V_{2}=(120 + 144)\pi\approx264\times3.14 = 829.0cm^{3}\).
• Answer: \(829.0cm^{3}\)

1. Question 5: Volume of a composite figure (rectangular prism with a smaller rectangular - prism removed)

• Explanation:
• Step 1: Calculate the volume of the larger rectangular prism
• The larger rectangular prism has dimensions \(l = 7.3m\), \(w = 4.6m\), \(h = 6.8m\). The volume \(V_{1}=lwh=7.3\times4.6\times6.8 = 229.784m^{3}\).
• Step 2: Calculate the volume of the smaller rectangular prism
• The smaller rectangular prism has dimensions \(l = 1m\), \(w = 1m\), \(h = 6.8m\). The volume \(V_{2}=1\times1\times6.8 = 6.8m^{3}\).
• Step 3: Calculate the total volume
• The total volume \(V=V_{1}-V_{2}=229.784 - 6.8=222.984\approx223.0m^{3}\).
• Answer: \(223.0m^{3}\)

1. Question 6: Volume of a composite figure (hemisphere and cylinder)

• Explanation:
• Step 1: Calculate the volume of the hemisphere
• The radius of the hemisphere \(r = 6in\). The volume of a hemisphere \(V_{1}=\frac{2}{3}\pi r^{3}=\frac{2}{3}\pi\times6^{3}=144\pi in^{3}\).
• Step 2: Calculate the volume of the cylinder
• The cylinder has radius \(r = 1.5in\) and height \(h = 5.8in\). The volume of a cylinder \(V_{2}=\pi r^{2}h=\pi\times(1.5)^{2}\times5.8 = 13.05\pi in^{3}\).
• Step 3: Calculate the total volume
• The total volume \(V=V_{1}+V_{2}=(144 + 13.05)\pi\approx157.05\times3.14 = 493.1in^{3}\).
• Answer: \(493.1in^{3}\)