Question

If the radii of the circular ends of a bucket that is 24 cm high are 5 cm and 15 cm, respectively, show that the surface area of the bucket is (545π) cm².

Answer 1

Let R and r be the radius of the top and base, respectively, of the bucket and let h be its height.

Then, R = 15 cm, r = 5 cm and h = 24 cm

Slant height, $l=\sqrt{h^2+{\left(R-r\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(24\right)}^2+{\left(15-5\right)}^2} \\ =\sqrt{576+100} \\ =\sqrt{676} \\ =26\text{cm} \end{gathered}$$

Surface area of the bucket$=\mathrm{\pi }\left[r^2+l\left(R+r\right)\right]$
$$\begin{gathered} =\mathrm{\pi }\times \left[{\left(5\right)}^2+26\times \left(15+5\right)\right] \\ =\mathrm{\pi }\times \left[25+520\right] \\ =545\mathrm{\pi }{\mathrm{cm}}^2 \end{gathered}$$