Question

One cylindrical barrel is placed inside another barrel, both of a height of 8.2 feet. The diameter of the smaller barrel is 1/3 the diameter of the larger one. If the inner barrel is left empty, and the diameter of the outer barrel is 6.9 feet, how much water is needed to fill the outer barrel? Use 3.13 for π.

Answer 1

Given:

• One cylindrical barrel is placed inside another barrel.
• Height of both the barrels = $8.2feet$
• Diameter of the large barrel = $6.9feet$
• Diameter of the small barrel = $(1/3)$ diameter of the large barrel

To find:

The volume of water needed to fill the outer barrel, keeping the inner barrel empty.

Note:

An image is attached with the solution, showing the arrangement, for better understanding.

To be recollected:

The conversion factors to be used in the calculations are:

$1 cubicfeet=0.0283cubic meters\\1cubic meter=1000litres$

Explanation:

• Here, the volume of water required to fill the outer barrel is to be calculated.
• Given that the barrels are of cylinder shape.
• This means that, the amount of water required to fill the outer cylinder is to be found out, which is the volume of the cylinder.
• Hence, the formula for the volume of cylinder = $\pi r^{2}h$
• If two cylinders are placed, one in another, and only the outer cylinder is to be used, then the volume of the portion in-between them is given by:

$V=\pi (r_2^{2}-r_1^{2})h$.......(1)

Where,

$r_1$ = Radius of the inner barrel

$r_2$ = Radius of the outer barrel

$h$ = Height of the barrel

$V$ = Volume of water required

Solution:

Let the diameter of the outer barrel be denoted by "$d_2$".

Let the diameter of the inner barrel be denoted by "$d_1$".

Now, from the given data:

The diameter of inner barrel is $1/3$ of the diameter of the outer barrel.

⇒$d_1=(1/3)d_2$

Hence, the diameter of the inner barrel is given by:

$d_1=(1/3)*6.9\\d_1=2.3feet$

Now, the radius of outer and inner barrels are given by:

$r_1=d_1/2\\r_1= 2.3/2\\r_1=1.15feet\\r_2=d_2/2\\r_2=6.9/2\\r_2=3.45feet$

Using the above data, given data and the equation (1), the quantity of water required is calculated as follows:

$V=\pi (r_2^{2}-r_1^{2})h\\V=3.13*(3.45^{2}-1.15^{2})*8.2\\V=3.13*(11.9025-1.3225)*8.2\\V=3.13*10.58*8.2\\V=271.54628cubic feet$

⇒$V=(271.54628)(0.0283)$

$V=7.685cubic metres\\V=7.685(1000)\\V=7685litres$

Final Answer:

The required amount of water to fill the outer barrel is:$271.54628cubic feet=7685litres$