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 28 cm and 44 cm? The hat side length is 30 cm. Add 5% of the material to the bust. Round to cm².

Final Answer:

S =  2335 cm²

Step-by-step explanation:

$$\begin{gathered} D_1=28\text{cm} \\ {D}_2=44\text{cm} \\ s=30\text{cm} \\ q=1+\frac{5}{100}=\frac{1\cdot 100}{100}+\frac{5}{100}=\frac{100}{100}+\frac{5}{100}=\frac{100+5}{100}=\frac{105}{100}=\frac{21}{20}=1.05 \\ {r}_1=D_1/2=28/2=14\text{cm} \\ {r}_2=D_2/2=44/2=22\text{cm} \\ {S}_1=\pi \cdot r_2^2-\pi \cdot r_1^2=3.1416\cdot 22^2-3.1416\cdot 14^2\doteq 904.7787{\text{cm}}^2 \\ {S}_2=\pi \cdot r_1\cdot s=3.1416\cdot 14\cdot 30\doteq 1319.4689{\text{cm}}^2 \\ S=q\cdot (S_1+S_2)=1.05\cdot (904.7787+1319.4689)=2335{\text{cm}}^2 \end{gathered}$$

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