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 $42$ $42$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $45 π$ $45 pi$ , 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 $42$ $42$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $45 π$ $45 pi$ , what is the area of the base of the cylinder?

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Answer 1 · 2018-03-19T11:03:20

$2.25 π$ $2.25pi$ units squared

Explanation:

To find the area, we first need to find the radius. We know that the total volume of the cylinder plus the cone is $45 π$ $45pi$ , which is found by adding the volumes of the cylinder ( $6 r^{2} π$ $6r^2pi$ ) and the cone ( $\frac{42 r^{2} π}{3}$ $(42r^2pi)/3$ ), where $r$ $r$ represents the unknown radius of the cone's and cylinder's bases.

Cylinder's volume: $π r^{2} \cdot h$ $pir^2*h$

Cone's volume: $\frac{π r^{2} \cdot h}{3}$ $(pir^2*h)/3$

Add the two volumes together:

$6 r^{2} π + \frac{42 r^{2} π}{3} = 45 π$ $6r^2pi+(42r^2pi)/3=45pi$

$\frac{18 r^{2} π}{3} + \frac{42 r^{2} π}{3} = 45 π$ $(18r^2pi)/3 +(42r^2pi)/3=45pi$

$\frac{60 r^{2} π}{3} = 45 π$ $(60r^2pi)/3=45pi$

$60 r^{2} π = 135 π$ $60r^2pi=135pi$

$60 r^{2} = 135$ $60r^2=135$

$r^{2} = 2.25$ $r^2=2.25$

$r = \pm 1.5 \to$ $r=+-1.5 rarr$ Disregard the -1.5, a length cannot be negative

Area is $π r^{2}$ $pir^2$

$π \cdot {1.5}^{2}$ $pi*1.5^2$

$π \cdot 2.25$ $pi*2.25$

The answer is $2.25 π$ $2.25pi$

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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 $42$ $42$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $45 π$ $45 pi$ , 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 42 4242 and the height of the cylinder is 6 66 . If the volume of the solid is 45 pi45π45 pi, what is the area of the base of the cylinder?

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Explanation:

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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 $42$ $42$ and the height of the cylinder is $6$ $6$ . If the volume of the solid is $45 π$ $45 pi$ , what is the area of the base of the cylinder?