Solve each problem. Volume of Water A water tank in the...

Solve each problem.
Volume of Water A water tank in the shape of an inverted cone has height 11 feet and radius 5.0 feet, as illustrated in the accompanying figure. Find the volume of the water in the tank when the water is 6 feet deep. (FIGURE CAN'T COPY)

Key Concepts

Volume of a Cone

This concept involves using the formula V = (1/3)?r²h to calculate the volume of any cone. In problems involving conical tanks, this formula is essential to find the volume of a partial or complete cone by substituting the appropriate radius and height.

Similar Triangles and Proportional Relationships

When a conical tank is partially filled, the surface of the water forms a smaller cone that is similar to the larger tank. By using the ratios provided by similar triangles, one can relate the water’s depth to the surface radius, allowing for the correct substitution into the volume formula for a cone.

Transcript

00:01 We have problem number 45 in which there is a water tank in conical shape whose height is height of the cone is 11 feet and radius is 5 feet now we have to find the volume of the water in the tank when the water is 6 feet deep so water is 6 feet deep so from distance here to here becomes 6 feet so the height of the water will be 11 minus 6 that is 5 feet a volume of water will be pyre square h by 3 pi r square h by 3 or 1 by 3 pyre square h so till now we have h equal to 5 feet we need to find the radius here okay now to get the this radius let us say r1 or r we have to just use the the fundamental of similar triangles in triangle this bigger triangle and this smaller triangle so these two triangles will be similar triangles as this angle is common in both the this angle is common in both the triangles that is triangle o -a -b and what we say o -a -b and o -a -b and o -m -n so in triangle o -a -b and o -m -n so in triangle o -a -b and o -m -n angle k -o -a -b a -b -b -b -a -b -a -b is congruent to angle m b n and one more angle will be the congruent as these two are parallel and this will be the corresponding angle so angle oab oab will be congruent to angle n m b so triangle oab will be similar to triangle o m b will be similar to triangle o m n by angle angle criteria now if this is similar so the sides will be proportion size will be in proportion so we should write oh a by an m o a by okay om n m m b n i have written wrong triangle i think here oab and nmb nm b n m b okay triangle n m b yes sides will be in proportion so o a by nm will be be o b by n b will be o b by n b okay so o a we have 5 that is given radius is 5 and m is r equal to ob is 11 feet and b is 5 okay so r will be equal to 25 by 11 feet now volume will become 1 by 3 pi r square that is 25 by 11 whole square into h that is 5 so this will be 1 by 3 into pi into 625 by 121 into 5.

04:51 Now we have to use the calculator for this.

04:53 This will be 27.

05:02 So volume of water will approximately be 27...

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