Question

If a spherical tank 6m in diameter can be filled with a liquid for $800, how do you find the cost to fill a tank 18m in diameter ?

Answer 1

Hint: In order to solve the question, first we will need to calculate the volume of both the spherical tanks having diameter of 6m and 18m by using the formula of volume of a sphere i.e. \[\dfrac{4}{3}\pi {{r}^{3}}\]. Then we will observe the relation between the volumes of both the tanks such that volume of tank 18m in diameter is how many times greater than the volume of a tank 6m in diameter. Later the same we will apply in the cost and we will get the required cost to fill the tank having 18m of diameter.

Complete step by step solution:
Volume of a sphere = \[\dfrac{4}{3}\pi {{r}^{3}}\] where r is the radius of the given spherical tank. We have given that,a spherical tank of 6m in diameter.Thus,volume of the spherical tank = \[\dfrac{4}{3}\pi {{r}^{3}}\]. As we know that diameter is the double or 2 times of radius. So diameter = 6m, then radius = 3m. Therefore,
volume of the spherical tank = \[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{4}{3}\pi \times {{\left( 3 \right)}^{3}}=\dfrac{4}{3}\pi \times 27=36\pi \]
Hence, the volume of a spherical tank of diameter 6m is \[36\pi \].

A spherical tank of 18m in diameter.Thus,volume of the spherical tank = \[\dfrac{4}{3}\pi {{r}^{3}}\].As we know that diameter is the double or 2 times of radius.So diameter = 18m, then radius = 9m. Therefore,
volume of the spherical tank = \[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{4}{3}\pi \times {{\left( 9 \right)}^{3}}=\dfrac{4}{3}\pi \times 729=972\pi \]
Hence, volume of a spherical tank of diameter 18m is \[972\pi \].

As we can observe that,the volume of a spherical tank of 18m in diameter is greater than the volume of a spherical tank having a diameter of 6m.Thus,volume of a spherical tank 18m in diameter = 27 times of the volume of spherical tank 6m in diameter
\[\Rightarrow 972\pi =27\times 36\pi \]
Therefore,the cost of filling a spherical tank 6m in diameter = $ 800$.Thus,the cost of filling a spherical tank 18m in diameter is also 27 times greater than the cost of filling a spherical tank 6m in diameter.So,cost of filling a spherical tank of 18m in diameter = \[27\times \$800=\$21600\].

Therefore, the cost to fill a tank 18m in diameter is $ 21600$.

Note:While solving these types of questions, students need to remember that they do not need to put the value of \[\pi \] during the calculating of the volume because it will only give you the volume in decimal form and it will complicate the answer for finding the cost. So keep the volume in terms of \[\pi \]as you will solve the question easily and you will definitely get the correct answer.