Question

Calculate the volume of the frustum cone given below with \[h = 15cm, r = 1.5cm\] and \[R = 15cm\] : (Use \[\pi = 3.14\] )
A. $3821.075\;c{m^3}$
B. $3921.075\;c{m^3}$
C. $3021.075\;c{m^3}$
D. $3121.075\;c{m^3}$

Answer 1

Hint:
We know That volume of the frustum of a cone when h,r and R is given is
 \[{V_{frustum}} = \dfrac{1}{3} \times \pi \times h \times ({R^2}\; + R \times r + {r^2})\]
Here, we will put the value of \[\pi \] as 3.14 for the ease of calculation.
Hence we will simply put the given values in a formula to get the final answer.

Complete step by step solution:

We know that the volume (V) of the frustum of the cone is given by
 \[{V_{frustum}} = \dfrac{1}{3} \times \pi \times h \times ({R^2}\; + R \times r + {r^2})\] … (1)

Where,
 \[h \to Height\,\,of\,\,the\,\,frustum\]
 \[R \to Radius\,\,of\,\,the\,\,bigger\,\,circular\,\,end\,\,the\,\,frustum\]
 \[r \to Radius\,\,of\,\,the\,\,smaller\,\,circular\,\,end\,\,the\,\,frustum\]
 \[\pi = 3.14\]

According to the given question
 \[ \Rightarrow h = 15cm\]
 \[ \Rightarrow R = 15cm\]
 \[ \Rightarrow r = 1.5cm\]

Hence, put \[h = 15cm\], \[R = 15cm\] and \[r = 1.5cm\] in (1), we get
 \[ \Rightarrow {V_{frustum}} = \dfrac{1}{3} \times \pi \times 15 \times ({15^2}\; + 15 \times (1.5) + {(1.5)^2})\]
On simplification we get,
 \[ \Rightarrow {V_{frustum}} = \dfrac{1}{3} \times \pi \times 15 \times (225 + 22.5 + 2.25)\]

The BODMAS rule states we should calculate the Brackets first, then the Order, then any Division or Multiplication and finally any Addition or Subtraction

According to BODMAS rule, we need to solve the brackets first, hence we get
 \[ \Rightarrow {V_{frustum}} = \dfrac{1}{3} \times \pi \times 15 \times (249.75)\]

On multiplication we get,
 \[ \Rightarrow {V_{frustum}} = \pi \times 1248.75\] … (2)

Now , put \[\pi = 3.14\]in (2), we get
 \[ \Rightarrow {V_{frustum}} = 3.14 \times 1248.75\]

Hence, The volume of the frustum of the cone is
 \[ \Rightarrow {V_{frustum}} = 3921.075c{m^3}\]

Hence, B is the final answer.

Note:
Be careful while doing these types of direct formula based questions. They may look very easy but different variables given in the question may have different units and hence they are first required to be converted into one common unit so that it can be solved. If not converted, the answer would surely be wrong. Luckily in the above question we had all our units in cm(centimeter) , That is why no unit conversions are done in the beginning of the question. Otherwise, unit conversion should be the first and the most important step in these types of questions.