Math CentralQuandaries & Queries


Question from jagjeet, a student:

dear suppose i have cone of 50cm radius at base and 100cm radius at top at a height of 80cm. now keeping base radius of 50cm and height of 80cm i want to increase top radius to 150cm how will i do that. plz height m talking abt is vertical height not slant height


I am going to use the diagram and notation that Steve used in his resonse to a similar question. In your case $t = 2 \times$ 50 cm, $b = 2 \times$ 150 cm and $h = 80$ cm. Thus $w^2$ = $80 ^2 + 100^2$ and hence $w = 128.06$ cm. Furthermore $r = 2 \times$ 50 $\pi$ = 100 $\pi$ and $R = 2 \times$ 150 $\pi$ = 300 $\pi$ and thus $\frac{r}{R}$ = $\frac{1}{3}$. Substituting $R = r + 128.06$ and solving for $r$ yields $r = 64.03$ cm and hence $R = 192.09$ cm.

Looking at the cone pattern is Steve's solution the full circumference of the inner circle is $2 \pi r = 128.06 \pi$ cm and the arclength is $100 \pi$ cm and $\frac{100 \pi}{128.06 \pi}$ = 78%. Thus the measure of the central angle is 78% of 360o = 218.1o.

I hope this helps,

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