Question

A bucket made up of metal sheet is in the form of a frustum of a cone. It's depth is $24cm$ and the diameters of the top and bottom are $30cm$ and $10cm$ respectively. Find the cost of milk which can completely fill the bucket at the rate of$Rs20$ per litre and the cost of the metal sheet used, if it costs$Rs10$ per $100c{m}^2$ (use $\mathrm{\pi }=3.14$ )

Answer 1

Step 1: Determine the capacity of the frustum :

The bucket has a larger diameter $=30cm$

The radius of the bigger end of the bucket:

$$\begin{gathered} \left(R\right)=\frac{diameter}{2} \\ =\frac{30}{2} \\ =15cm \end{gathered}$$

The bucket's smaller end's diameter $=10cm$

The radius of the smaller end of the bucket :

$$\begin{gathered} \left(r\right)=\frac{10}{2} \\ =5cm \end{gathered}$$

The height of the bucket $=24cm$

The slant height $\left(l\right)$ :

$$\begin{gathered} \sqrt{h^2+{(R-r)}^2} \\ =\sqrt{{24}^2+{(15-5)}^2} \\ =\sqrt{576+100} \\ =\sqrt{676} \\ =26cm \end{gathered}$$

The capacity of the frustum :

$$\begin{gathered} =\frac{1}{3}\mathrm{\pi h}({\mathrm{R}}^2+{\mathrm{r}}^2+\mathrm{Rr}) \\ =\frac{1}{3}\times 3.14\times 24({15}^2+5^2+15\times 5) \\ =3.14\times 8(225+25+75) \\ =3.14\times 8\times 325 \\ =8164{\mathrm{cm}}^3 \\ =8.164\mathrm{litres} \end{gathered}$$

Step 2: Determine the cost of the metal used for making the bucket :

A liter of milk cost is $Rs20$

The total cost of filling the bucket with milk $$\begin{gathered} =8.164\times 20 \\ =Rs163.28 \end{gathered}$$

The surface area of the bucket :

$$\begin{gathered} {\mathrm{\pi r}}^2+\mathrm{\pi l}(\mathrm{R}+\mathrm{r}) \\ =\mathrm{\pi }\left[{\mathrm{r}}^2+\mathrm{l}(\mathrm{R}+\mathrm{r})\right] \\ =3.14\left[5^2+26(15+5)\right] \\ =3.14\left[25+26\times 20\right] \\ =3.14\left[25+520\right] \\ =1711.3{\mathrm{cm}}^2 \end{gathered}$$
The cost $100c{m}^2$ of the metal sheet is $Rs10$

The cost of the metal used for making the bucket :$$\begin{gathered} \frac{1711.3}{100}\times 10 \\ =Rs171.13 \end{gathered}$$
Hence, the cost of the metal used for making the bucket is $Rs171.13$