Question

A bucket is in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the capacity and surface area of the bucket. Also, find the cost of milk which can completely fill the container , at thr rate of ₹25 per litre. (Use $\mathrm{\pi }=3.14).$

Answer 1

Height of the bucket, h = 30 cm
Radii r₁ = 10 cm and r₂ = 20 cm
Capacity of the bucket,
$$\begin{gathered} V=\frac{1}{3}\mathrm{\pi h}\left({\mathrm{r}}_1^2+{\mathrm{r}}_1{\mathrm{r}}_2+{\mathrm{r}}_2^2\right) \\ =\frac{1}{3}\mathrm{\pi }\times 30\left({10}^2+10\times 20+{20}^2\right) \\ =21980{\mathrm{cm}}^3 \\ =21.980\mathrm{liters} \end{gathered}$$

$$\begin{gathered} l=\sqrt{h^2+{\left(r_2-r_1\right)}^2} \\ l=\sqrt{{30}^2+{\left(20-10\right)}^2}=10\sqrt{10} \end{gathered}$$
Surface area of the bucket
$$\begin{gathered} S=CSA+areaofthebase \\ S=\mathrm{\pi }\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)\mathrm{l}+{{\mathrm{\pi r}}_1}^2 \\ \mathrm{S}=\mathrm{\pi }\left(10+20\right)10\sqrt{10}+\mathrm{\pi }{\left(10\right)}^2 \\ \mathrm{S}=2978.86+314=3292.86{\mathrm{cm}}^2 \end{gathered}$$
Cost of milk which can completely fill the container at Rs 25/litre
= 21.980 × 25
= Rs 549.50