Question

If the radii of the circular ends of a conical bucket which is \[45\]cm high be \[28\]cm and \[7\]cm, find the capacity of the bucket.

Answer 1

Hint: The question requires finding the capacity of the conical bucket which is the same as finding volume of the frustum of cone.
A frustum is the part of a solid that lies between one or two parallel planes that break it in two parts in geometry. Meanwhile the radii are the radius of the circular part of the cone. We can use formula to find the volume since we have information about height and radii.

Complete step-by-step answer:
We are given the radii of the circular ends of a conical bucket which is \[45\]cm high be \[28\]cm and \[7\]cm.
Now, we have to find the capacity of the bucket.
The formula to find volume of the frustum of cone is:
\[ = \dfrac{1}{3}\pi h(r_1^2 + r_2^2 + {r_1}{r_2})\]
In the given case, we are provided with height and radii i.e.
\[h = 45cm\], \[{r_1} = 28cm\]and \[{r_2} = 7cm\]
Comparing the given information with the formula, we get,
\[ = \dfrac{1}{3}\pi (45)({(28)^2} + {(7)^2} + (28)(7))\]
Substituting the value of \[\pi = \dfrac{{22}}{7}\], we get,
\[ = \dfrac{1}{3}(\dfrac{{22}}{7})(45)({(28)^2} + {(7)^2} + (28)(7))\]
Solving the bracket, we get,
\[ = \dfrac{1}{3}(\dfrac{{22}}{7})(45)(784 + 49 + 196)\]
\[ = \dfrac{1}{3}(\dfrac{{22}}{7})(45)(1029)\]
Dividing by the denominators, we get,
\[ = 1 \times 22 \times 15 \times 147\]
Solving the multiplication, we get,
\[ = 48510\]
Since the data is given in centimetres and we are finding the volume of frustum of cone, the unit of measurement will be \[c{m^3}\]. Hence final answer will be denoted as:
\[ = 48510c{m^3}\]
Therefore, the capacity of the conical bucket is \[48510c{m^3}\].

Note: A conical bucket has two radii since there are two circles. The diagrammatic representation of the given sum is shown as follows:

Frustum is the space between the two circles i.e. space created by cutting the cone with two parallel lines. Other than bucket, examples of frustum of cone are coffee mug, plant pot, cupcake, conical container for storage of goods etc.