Question

The diameter and height of a circular cylinder are 20 feet and 4 feet respectively. What is the volume of water in gallons, if the given cylinder is filled to a height of 3 feet and 6 inches? ( 1 feet \[ = \]12 inches, 1 gallon \[ = \]231 cubic inches)
A.\[\dfrac{{56700}}{7}\]
B. \[\dfrac{{57600}}{7}\]
C. \[\dfrac{{58600}}{7}\]
D. \[\dfrac{{58700}}{7}\]
E. \[\dfrac{{59600}}{7}\]

Answer 1

Hint: We convert all measurements of the circular cylinder from feet to inches. Use the height up to which water is filled and radius half of the diameter and calculate the volume of water filled in the cylinder using the formula of volume of a cylinder. Convert the volume obtained from cubic inches to gallons using the given conversion
* A cylinder having radius of the base as ‘r’ and height ‘h’ has volume \[\pi {r^2}h\]
* Radius of a circle is half the value of the diameter.
* 1 feet \[ = \]12 inches, 1 gallon \[ = \]231 cubic inches

Complete step-by-step solution:
We have a cylinder of height 4 feet and diameter of the base as 20 feet
Since we have to fill the cylinder up to 3 feet and 6 inches, we take the height ‘h’ as 3 feet 6 inches.
Since we are given diameter of the base as 20 feet
\[ \Rightarrow \]radius of the base of the cylinder \[ = \dfrac{{20}}{2}\]
\[ \Rightarrow \]radius of the base of the cylinder \[ = 10\]feet
Now we convert the value of radius and height from feet to inches.
Since 1 feet \[ = 12\]inches
\[ \Rightarrow \]10 feet \[ = 12 \times 10\]inches
\[ \Rightarrow \]10 feet \[ = 120\]inches
\[ \Rightarrow \]Radius of the base, \[r = 120\]inches … (2)
Now we have height of cylinder as 3 feet and 6 inches
Since 1 feet \[ = 12\]inches
\[ \Rightarrow \]3 feet \[ = 12 \times 3\]inches
\[ \Rightarrow \]3 feet \[ = 36\]inches
So, 3 feet 6 inches will be \[ = 36 + 6\]inches
\[ \Rightarrow \]3 feet 6 inches \[ = 42\]inches
\[ \Rightarrow \]Height of the cylinder, \[h = 42\]inches … (3)
Since we know volume of a cylinder is given by \[\pi {r^2}h\]
Substitute the values from equations (2) and (3) in the formula of volume
\[ \Rightarrow V = \left( {\dfrac{{22}}{7} \times 120 \times 120 \times 42} \right)\]cubic inches
Cancel same factors from numerator and denominator
\[ \Rightarrow V = \left( {22 \times 120 \times 120 \times 6} \right)\]cubic inches
\[ \Rightarrow V = 1900800\]cubic inches
Since we need volume in terms of gallons, we use the conversion of cubic inches into gallons i.e. 1 gallon \[ = \]231 cubic inches
Since, 231 cubic inches \[ = \] 1 gallon
\[ \Rightarrow \]1 cubic inch \[ = \dfrac{1}{{231}}\]gallon
\[ \Rightarrow \]1900800 cubic inch \[ = \dfrac{1}{{231}} \times 1900800\]gallon
Cancel same factors from numerator and denominator
\[ \Rightarrow \]1900800 cubic inch \[ = \dfrac{{57600}}{7}\]gallon
\[\therefore \]Volume of water is \[\dfrac{{57600}}{7}\]gallon

\[\therefore \]Option A is correct.

Note: Many students make mistake of calculating the volume in terms of cubic feet and then use conversion given on internet to convert the answer into gallons, keep in mind we are given a certain conversion so we make use of it, also if we convert from feet to gallons we will not be able to get answer matching the options.