Question

A quarter circular sector is removed from a circle and the remainder is folded into a cone by connecting the
cut edges. When viewed from the side, what is the angle at the apex of the cone?
How do you solve this trigonometry problem?

Answer 1

Answer:

Use Case 9 since we know the radius, s = 24 inches and we are solving for the central angle, T = θ, which will maximize the volume, V = πr2h/3.

 

From Case 9, we know for the cone, r = sT/(2π) and h = √[s2 - r2].

 

Therefore, V = (π/3)[(sT)/(2π)]2(√[s2 - ((sT)/(2π))2])

 

To maximize V find dV/dT and set it equal to zero.

 

Differentiate using the product rule, resulting ultimately after simplification (and substituting θ for T) as:

 

(2θs3/4π)√[(4π2-θ2)/(4π2)] - (θ3s3)/(4π√[(4π2-θ2)/(4π2)]) = 0

 

So, (2θs3/4π)√[(4π2-θ2)/(4π2)] = (θ3s3)/(4π√[(4π2-θ2)/(4π2)])

 

θ2 = 4π2/(4π2+1) θ = √[4π2/(4π2+1)] θ = 0.98757049215 radians θ = 56.5836211719 degrees