Cap

Jesters hat is shaped by a rotating cone. Calculate how much paper is needed to the cap 54 cm high when the head circumference is 47 cm.

Correct result:

S =  1269 cm2

Solution:

v=54 cm o=47 cm  o=2 πr  r=o2π=472 3.14167.4803 cm  s=r12+v2=r12+542=54 cm  S=π r s=3.1416 7.4803 54=1269 cm2v=54 \ \text{cm} \ \\ o=47 \ \text{cm} \ \\ \ \\ o=2 \ \pi r \ \\ \ \\ r=\dfrac{ o }{ 2 \pi }=\dfrac{ 47 }{ 2 \cdot \ 3.1416 } \doteq 7.4803 \ \text{cm} \ \\ \ \\ s=\sqrt{ r_{1}^2 + v^2 }=\sqrt{ r_{1}^2 + 54^2 }=54 \ \text{cm} \ \\ \ \\ S=\pi \cdot \ r \cdot \ s=3.1416 \cdot \ 7.4803 \cdot \ 54=1269 \ \text{cm}^2



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