A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is $12$ $12$ and the height of the cylinder is $24$ $24$ . If the volume of the solid is $42 π$ $42 pi$ , what is the area of the base of the cylinder?

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A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is $12$ $12$ and the height of the cylinder is $24$ $24$ . If the volume of the solid is $42 π$ $42 pi$ , what is the area of the base of the cylinder?

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Answer 1 · 2018-08-04T01:37:26

$Cylinder base area = A = \frac{3}{2} π sq units$ $color(maroon)("Cylinder base area " = A = 3/2 pi " sq units"$

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$Volume of cone V_{c o \neq} = \frac{1}{3} π r^{2} h_{1}$ $"Volume of cone " V_(cone) = 1/3 pi r^2 h_1$

$Volume of cylinder V_{c y l} = π r^{2} h_{2}$ $"Volume of cylinder " V_(cyl) = pi r^2 h_2$

$Volume of solid V = \frac{1}{3} π r^{2} h_{1} + π r^{2} h_{2} = π r^{2} (\frac{1}{3} h_{1} + h_{2})$ $"Volume of solid " V = 1/3 pi r^2h_1 + pi r^2 h_2 = pi r^2 (1/3 h_1 + h_2)$

$Area of cylinder base = A = π r^{2}$ $"Area of cylinder base " = A = pi r^2$

$Given V = 42 π, h_{1} = 12, h_{2} = 24, π r^{2} = ?$ $"Given " V = 42 pi, h_1 = 12, h_2 = 24, pi r^2 = ?$

$π r^{2} (\frac{1}{3} h_{1} + h_{2}) = 42 π$ $pi r^2 (1/3 h_1 + h_2) = 42 pi$

$π r^{2} = \frac{42 π}{\frac{1}{3} h_{1} + h_{2}}$ $pi r^2 = (42 pi) / (1/3 h_1 + h_2)$

$A = π r^{2} = \frac{42 π}{\frac{1}{3} \cdot 12 + 24} = \frac{42 π}{28}$ $A = pi r^2 = (42 pi) / (1/3 * 12 + 24) = (42 pi) / 28$

$A = \frac{3}{2} π sq units$ $color(maroon)(A = 3/2 pi " sq units"$

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A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is $12$ $12$ and the height of the cylinder is $24$ $24$ . If the volume of the solid is $42 π$ $42 pi$ , what is the area of the base of the cylinder?

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A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is 12 1212 and the height of the cylinder is 24 2424 . If the volume of the solid is 42 pi42π42 pi, what is the area of the base of the cylinder?

1 Answer

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A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is $12$ $12$ and the height of the cylinder is $24$ $24$ . If the volume of the solid is $42 π$ $42 pi$ , what is the area of the base of the cylinder?