A conical drinking cup is made from a circular piece of paper of radius R=4 cm with center C by cutting out a sector and joining the edges CA and CB. Determine the maximum capacity of such a cup.

Question

Can Someone answer this without making the claim that R is the slant of a the cone and not the base? or if you do please explain the logic being why it would be the slant and not the base as in the picture which is initially a 2d circle the R or radius is obviously on the base and represents the half of the base of the cone. so when we turn it upright so we can see the cone Radius becomes the slant??? I don't see the logic or intuition there? But wee aren't changing the structural being of the object because we cant so why would the base which in my eyes has to be the Radius R be the slant length???

Answer 1

Let θ is central angle of this sector. Then 2πr = 4θ, where r is radius of cup's base. Then r = 2θ/π cm.

Height of cup h = √16 - 4θ²/π² = 2√4 - θ²/π².

So, volume v = πr²h/3 = π·4/3θ²/π²·2√4 - θ²/π² = 8/3θ²/π√4 - θ²/π².

v' = 8/(3π)[2θ√4 - θ²/π² - θ³/(π²√4 - θ²/π²)) = 8/(3π)(8θ - 3θ³)/(π²√4 - θ²/π²) = 0; 8θ - 3θ^√3= 0; θ = 0; θ = √8/3 = 2√6/3. At θ = 0 v = 0 (min).

At θ = 2√6/3 v= 8/3·8/3√4 - 8/(3π²)·1/π = 4.3715 cm³