Question

The diameters of the lower and upper end of a bucket in the form of a frustum of cone are 10 cm and 30 cm respectively. If its height is 24 cm. Find the area of the metal sheet used to make the bucket.

Answer 1

Hint: First, we will use the formula of the slant height \[l = \sqrt {{h^2} + {{\left( {{r_2} - {r_1}} \right)}^2}} \] and the total surface area of a frustum is \[\pi \left( {{r_1} + {r_2}} \right)l + \pi {r_1}^2\], where \[l\] is the length of the frustum and \[{r_1}\]and \[{r_2}\] are the radii.

Complete step-by-step answer:
Apply these formulae, and then use the given conditions to find the required value.
Given that the height of the frustum \[h\] is 24 cm and the diameters of the lower end \[{d_1}\] and upper end \[{d_2}\] of a bucket in the form of a frustum of cone are 10 cm and 30 cm respectively.
Let the radii of the lower and upper end of a bucket are \[{r_1}\] and \[{r_2}\] respectively.
Finding the value of the radii \[{r_1}\] and \[{r_2}\] from the given diameters of the lower and upper end of a bucket.
\[
  {r_1} = \dfrac{{10}}{2} \\
   = 5{\text{ cm}} \\
\]
\[
  {r_2} = \dfrac{{30}}{2} \\
   = 15{\text{ cm}} \\
 \]
We know that the slant height \[l\] is calculated using the formula, \[l = \sqrt {{h^2} + {{\left( {{r_2} - {r_1}} \right)}^2}} \], where \[h\] is the height of the frustum and \[{r_1}\]and \[{r_2}\] are the radii.
Substituting the values of \[h\], \[{r_1}\] and \[{r_2}\] in the above formula for slant height, we get
\[
   \Rightarrow l = \sqrt {{{24}^2} + {{\left( {15 - 5} \right)}^2}} \\
   \Rightarrow l = \sqrt {{{24}^2} + {{10}^2}} \\
   \Rightarrow l = \sqrt {576 + 100} \\
   \Rightarrow l = \sqrt {676} \\
   \Rightarrow l = 26{\text{ cm}} \\
\]

We know that the area of the metal sheet used to make the bucket is equal to the total surface area of the bucket excluding the upper end.
We also know that the formula to calculate the total surface area of a frustum is \[\pi \left( {{r_1} + {r_2}} \right)l + \pi {r_1}^2\], where \[l\] is the length of the frustum and \[{r_1}\]and \[{r_2}\] are the radii.
Substituting these values of \[{r_1}\], \[{r_2}\] and \[l\] in the above formula for total surface area, we get
\[
  {\text{Total Surface Area}} = \left( {3.14} \right)\left( {15 + 5} \right)\left( {26} \right) + 3.14{\left( 5 \right)^2} \\
   = 3.14\left( {20} \right)\left( {26} \right) + 3.14\left( {25} \right) \\
   = 3.14\left( {520} \right) + 78.5 \\
   = 1632.8 + 78.5 \\
   = 1711.3{\text{ c}}{{\text{m}}^2} \\
\]
Thus, the total surface area of the frustum is \[1711.3\] cm\[^2\].
Therefore, the area of the metal sheet used to make the bucket is \[1711.3\] cm\[^2\].

Note: In solving these types of questions, you should be familiar with the formula of slant height and the total surface area of the cone. Then use the given conditions and values given in the question, and substitute the values in the formula, to find the required value. Also, we are supposed to write the values properly to avoid any miscalculation.