Question

A bucket is in the shape of a frustum with the top and bottom circles of radii \[15\] cm and \[10\] cm. its depth is \[12\] cm. Find its curved surface area and total surface area. (Express the answer in terms of \[\pi \])

Answer 1

Hint: Firstly we find the slant height. After that we will substitute that value in the CSA and TSA formula to find the $CSA$ & $TSA$.

Formula used: Using these formulas we will find the curved and total surface area of the given shape.
The curved surface area of the frustum is $CSA$ = \[\pi (R + r)s = \pi (R + r)\sqrt {{{(R - r)}^2} + {h^2}} \]
And the total surface area is $TSA$ = \[\pi (R + r)s + \pi {r^2} + \pi {R^2} = \pi (R + r)\sqrt {{{(R - r)}^2} + {h^2}} + \pi ({r^2} + {R^2})\]

Complete step-by-step answer:
Let us consider, $R$, $r$, $s$ & $h$ be the radius of the lower base, radius of the upper base, the slant height and the perpendicular height of the frustum.
It is given that a bucket is in the shape of a frustum with the top and bottom circles of radii \[15\] cm and \[10\] cm and its depth is \[12\] cm. We have to find its curved surface area and total surface area.
Now, substitute, \[R\] = 15, \[r\] = 10 and \[h\] = 12 in lateral surface area we get,
 The lateral surface area \[CSA\] = \[\pi (15 + 10)\sqrt {{{(15 - 10)}^2} + {{12}^2}} \] ${cm^2}$
By simplifying the squares and square roots we get,
The lateral surface area \[CSA\] = \[\pi \times 25 \times 13\] ${cm^2}$
And on further simplifications we get,
The lateral surface area of the bucket \[CSA\] = \[ 325 \pi \] ${cm^2}$
Again, let us substitute, \[R\] = 15, \[r\] = 10 and \[h\] = 12 in total surface area we get,
 The total surface area $TSA$ = \[\pi (15 + 10)\sqrt {{{(15 - 10)}^2} + {{12}^2}} + \pi ({10^2} + {15^2})\] ${cm^2}$
By simplifying the squares and square roots we get,
The total surface area $TSA$ = \[\pi \times 25 \times 13 + 325\pi \] ${cm^2}$
And on further simplification we get,
The total surface area of the bucket $TSA$ = \[650 \pi \] ${cm^2}$
Hence, the curved surface area is \[325\pi \] ${cm^2}$ and total surface area is \[650\pi \] ${cm^2}$

Note: The frustum is the sliced part of a right circular cone. If we eliminate the top corner part of the right circular cone we get a frustum.