Fine the surface area of the ice cream cone shape by combining the surface area of the cone with the surface area of the hemisphere.
We can find the slanted length of the cone by using the pythagorean theorem. The slant is the hypoteneuse and the other sides of the triangle are the radius and height.
\( l^2 = h^2 + r^2 = h^2 + \cfrac{d}{2}^2 \)
\(l = \sqrt{(14)^2+(\cfrac{16}{2})^2} \)
\(l = 16.12 \; \text{in}\)
The surface area of a cone is:
\(SA = \pi r (l + r)\)
\(SA_{cone} = \pi \cfrac{16}{2} ( 16.12 + \cfrac{16}{2})\)
\(SA_{cone} = \pi (8) (24.12)\)
\(SA_{cone} = 606.20 \; \text{in}^2\)
The surface area of the hemisphere is half a sphere:
\(SA = \cfrac{1}{2} 4 \pi r^2 \)
\(SA_{hemisphere} = 2 \pi (\cfrac{16}{2})^2\)
\(SA_{hemisphere} = 402.12 \; \text{in}^2 \)
The surface area is:
\(SA = SA_{cone} + SA_{hemisphere} \)
\(SA = 606.2 + 402.12 = 1008.32 \; \text{in}^2\)
The surface area is \(1008.32 \; \text{yd}^2 \).
Find the volume of the ice cream cone shape by adding the volume of the cone and hemisphere:
The volume of a cone is:
\(V = \cfrac{1}{3} \pi r^2 h\)
\(V_{cone} = \cfrac{1}{3} \pi \cfrac{16}{2}^2 (14)\)
\(V_{cone} = 938.29 \; \text{yd}^3\)
The volume of the hemisphere is half a sphere:
\(V = \cfrac{1}{2} \cfrac{4}{3} \pi r^3\)
\(V_{hemisphere} = \cfrac{2}{3} \pi (\cfrac{16}{2})^3\)
\(V_{hemisphere} = 1072.33 \; \text{yd}^3\)
The volume is the sum of the cone and hemisphere:
\(V = V_{cone} + V_{hemisphere}\)
\(V = 938.29 + 1072.33 \)
\(V = 2010.62 \; \text{yd}^3\)
The volume is \(1164 \; \text{yd}^3\).
