Question

A hemispherical tank is filled with water and has a diameter of $10$ feet. If water weighs $62.4$ pounds per cubic foot. What is the total weight of the water in a full tank to the nearest pound?

Answer 1

Hint: We know that the volume of a hemisphere is half that of a sphere. So, to find the volume of the hemisphere we divide the volume of a sphere by two. Also, to find the radius we need to divide the given diameter by two. We have to use the formula of volume of a sphere in this question. At last, we have to multiply the volume with the given weight to find the total weight of water.

Formula used:
Volume of sphere $ = \dfrac{4}{3}\pi {r^3}$

Complete step by step answer:
In the above question, it is given that the diameter of the hemispherical tank is $10$ feet.
Therefore, radius of hemisphere $ = $ radius of sphere $ = $ $\dfrac{{10}}{2}feet = 5feet$
We know that the volume of a hemisphere is half that of a sphere.
Volume of hemisphere $ = $ $\dfrac{1}{2}\, \times $ volume of sphere
$ \Rightarrow \dfrac{1}{2} \times \dfrac{4}{3}\pi {r^3}$
Now put the value of r in the above equation.
$ \Rightarrow \dfrac{2}{3}\pi {\left( 5 \right)^3}$
$ \Rightarrow \dfrac{2}{3} \times 3.14 \times 125$
$ \Rightarrow 392.5\,cubic\,feet$
It is given that the weight of a cubic foot is $62.4$ pounds.
Therefore, to find the weight of $392.5\,cubic\,feet$, we need to multiply it by $62.4$.
The total weight of the water in a full tank $ = 62.4 \times 392.5\,pounds$
$ \Rightarrow 24492\,pounds$

Note:The easiest way to approach these kinds of problems is to solve it as if the question were asking about a full sphere, then divide by 2 (since it is hemispherical). Also, we should know that if an amount is given for one unit, then we have to multiply it with the total number of quantities to find the total amount.