SOLUTION: a container is in shape of a frustum of a cone. its diameter at the bottom is 18 cm and at the top 30 cm. If the depth is 24 cm determine the capacity of the container, correct to

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Question 1203336: a container is in shape of a frustum of a cone. its diameter at the bottom is 18 cm and at the top 30 cm. If the depth is 24 cm determine the capacity of the container, correct to the nearest liter, by the prismoidal rule. (1 liter= 1000 cm³)
Answer by ikleyn(52800) About Me  (Show Source):
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a container is in shape of a frustum of a cone. its diameter at the bottom is 18 cm and at the top 30 cm.
If the depth is 24 cm determine the capacity of the container, correct to the nearest liter,
by the prismoidal rule. (1 liter= 1000 cm³)
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Make a sketch of the vertical section of the frustum.
Continue non-parallel sides of the frustum (in the section) to get an isosceles triangle.

On the sketch, find two similar isosceles triangles.

Their bases are 18 cm and 30 sm.
Their altitudes are h (for smaller triangle) and (h+24) cm.

From similarity, write a proportion

    h%2F%28h%2B24%29 = 18%2F30,

or

    h%2F%28h%2B24%29 = 3%2F5.


From the proportion,

    5h = 3(h+24)  --->  5h = 3h + 72  --->  2h = 72  --->  h = 72/2 = 36.


Thus the height of the smaller cone is 36 cm;  the height of the larger cone is  36+24 = 60 cm.


Volume of the small cone is  %281%2F3%29%2Api%2Ar%5E2%2Ah = %281%2F3%29%2Api%2A9%5E2%2A36 = 972%2Api cm^3.

Volume of the large cone is  %281%2F3%29%2Api%2AR%5E2%2A%28h%2B24%29 = %281%2F3%29%2Api%2A15%5E2%2A60 = 4500%2Api cm^3.


The volume of the frustrum is the difference of the volumes of cones

    The volume of the frustrum  = 4500%2Api-972%2Api = 4500%2A3.14159265-972%2A3.14159265 = 11083.53887 cm^3.


To get the volume in liters, divide this value by 1000

    The volume of the frustrum in liters = 11083.53887%2F1000 = 11.083 liters = 11 liters  (rounded).    ANSWER

Solved.