Question

A conical cup is made from a circular piece of paper with radius 4 cm by cutting out a sector and joining the edges as shown below. Suppose θ = 3π/2.Find the volume V of the cup.

Answer 1

Answer:

  V = 3π√7 cm³ ≈ 24.94 cm³

Step-by-step explanation:

The radius of the finished cone can be found from the circumference of the finished cone, which is 3/4 of the circumference of the original paper circle.

  Ccone = (3/4)Ccircle = (3/4)(2π(4 cm)) = 6π cm

This is related to the cone radius (x) by ...

  Ccone = 2πx = 6π cm

  x = 3 cm . . . . . . . . . 3/4 of the circle's radius

The height of the cone is given by the Pythagorean theorem:

  (4 cm)² = h² + x²

  h² = (4 cm)² -(3 cm)² = (16 -9) cm²

  h = √7 cm

Now, we can use the formula for the volume of a cone:

  V = (1/3)πr²h

  = (1/3)π(3 cm)²(√7 cm)

  V = 3π√7 cm³ ≈ 24.94 cm³