Masquerade ball

Marie wants to make a cone-shaped witch's hat for a masquerade ball. How much material will it need if it counts on an annular rim with diameters of 28cm and 44cm? Hat side length is 30cm. Add 5% of the material to the bust. Round to cm².

Correct answer:

S =  2335 cm²

Step-by-step explanation:

$$\begin{gathered} D_1=28\text{ cm } \\ {D}_2=44\text{ cm } \\ s=30\text{ cm } \\ q=1+{\displaystyle \frac{5}{100}}={\displaystyle \frac{21}{20}}=1.05 \\ {r}_1=D_1\mathrm{/}2=28\mathrm{/}2=14\text{ cm } \\ {r}_2=D_2\mathrm{/}2=44\mathrm{/}2=22\text{ cm } \\ {S}_1=\pi \cdot r_2^2-\pi \cdot r_1^2=3.1416\cdot 22^2-3.1416\cdot 14^2\doteq 904.7787{\text{cm}}^2 \\ {S}_2=\pi \cdot r_1\cdot s=3.1416\cdot 14\cdot 30\doteq 1319.4689{\text{cm}}^2 \\ S=q\cdot (S_1+S_2)=1.05\cdot (904.7787+1319.4689)=2335{\text{cm}}^2 \end{gathered}$$

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