We can make a skecth of the bead as:
Then, we can find the approximate area by listing all the areas that will be covered:
0. Area of the top (A1), that will be the area of the bigger circle minus the area of the smaller circle.
,1. Area of the base (A2), equal to the area of the top (we can add them when calculating).
,2. Lateral surface (A3) of the bigger cylinder.
,3. Lateral surface (A4) of the inner cylinder.
Then, we can write the area A as the sum of the areas we have just listed:
[tex]A=A_1+A_2+A_3+A_4[/tex]We can then write them in function of the radius and the height as
[tex]\begin{gathered} A=A_1+A_2+A_3+A_4 \\ A=\pi(R^2-r^2)+\pi(R^2-r^2)+2\pi Rh+2\pi rh \end{gathered}[/tex]We can group the terms and replace the variables with their values and then solve:
[tex]\begin{gathered} A=\pi(R^2-r^2)+\pi(R^2-r^2)+2\pi Rh+2\pi rh \\ A=2\pi(R^2-r^2)+2\pi h(R+r) \\ A=2\pi(1^2-0.5^2)+2\pi\cdot1.5\cdot(1+0.5) \\ A=2\pi(1-0.25)+2\pi\cdot1.5\cdot1.5 \\ A=2\pi\cdot0.75+2\pi\cdot2.25 \\ A=1.5\pi+4.5\pi \\ A=6\pi \\ A\approx18.85\operatorname{cm}^2 \end{gathered}[/tex]Answer: the approximate area is 18.85 cm² (exact solution: 6π cm²)
