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? The 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:

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

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