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Question 841527: A Tank have dimension 21 decimeter long ,11 decimeter wide, and 6 decimeter deep fill with half of water.A solid sphere have whose diameter is 21 centimeter, if we droped a 100 number of solid sphere into that tank then how much water level is rise in the tank
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first thing you want to do is get them into the same measurements.
since 1 decimeter is equal to 10 centimeters, then the dimensions of the box would be:
210 centimeters long
110 centimeters wide
60 centimeters deep.
the volume of the box is therefore equal to 210 * 110 * 60 cubic centimeters.
this is equal to 1386000 cubic centimeters.
the ratio of the volume to the depth is equal to 1386000 / 60 which is equal to 23100 square centimeters of volume per centimeter of height.
since the tank is half full, then the volume of water in the tank would be equal to 30 * 23100 which is equal to 693000 cubic centimeters.
take 1/2 of 1386000 and you get 693000 so this makes sense.
the formula for the volume of a sphere is equal to 4/3 * pi * r^3
if the diameter of a sphere is equal to 21 cm, then the radius is equal to half of that which makes the radius equal to 10.5 cm.
the volume of one of the spheres is equal to 4/3 * pi * (10.5)^3 which is equal to 4849.048261 cubic centimeters.
since there are 100 spheres, the total volume of all the spheres combined is equal to 484904.8261 cubic centimeters.
placing the 100 spheres in the tank displaces the same volume of water, so the total number of cubic centimeters of water in the tank becomes equal to 693000 plus 484904.8261 which is equal to 1177904.826 cubic centimeters.
since the ratio of cubic centimeters to height is equal to 231000 to 1, we can use this ratio to estimate the height of the water with all the spheres in there.
the ratio becomes:
23100 / 1 = 1177904.826 / x
cross multiply to get 23100 * x = 1177904.826
divide both sides of this equation by 23100 to get x = 50.99155091.
the height of the water in the tank will rise from 30 centimeters to 50.99155091 centimeters.
a confirmation check will show that:
height of the tank = 60 * 23100 = 1386000 cubic centimeters (good).
height of the water in the tank = 30 * 231000 = 693000 cubic centimeters (good).
volume of 100 spheres = 100 * 4/3 * pi * (10.5)^3 = 484904.8261 cubic centimeters (good).
volume of water in the tank is increased to 693000 + 484904.8261 cubic centimeters which is equal to 1177904.826 cubic centimeters (good).
using the ratio of 23100 square centimeters per centimeter of height, we get the new height is equal to 1177904.826 / 23100 = 50.99 centimeters rounded to 2 decimal places.
the height of the water level was raised from 30 to 50.99 which is an increase of 20.99 centimeters in height.
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