Math Problem Statement

Find the volume of the cone. Either enter an exact answer in terms of π or use 3.14 for π and round final answer to the nearest hundredth.

Solution

To find the volume of the cone, we use the formula:

V=13πr2hV = \frac{1}{3} \pi r^2 h

where:

  • rr is the radius of the cone’s base,
  • hh is the height of the cone.

From the image:

  • The radius r=3r = 3,
  • The height h=2h = 2.

Substitute these values into the formula:

V=13π(3)2(2)=13π×9×2=6πV = \frac{1}{3} \pi (3)^2 (2) = \frac{1}{3} \pi \times 9 \times 2 = 6 \pi

If you need an approximate answer using π3.14\pi \approx 3.14:

V6×3.14=18.84V \approx 6 \times 3.14 = 18.84

The volume of the cone is approximately 18.8418.84 cubic units, or exactly 6π6\pi cubic units.

Would you like further details on this solution?

Here are five questions to expand on this topic:

  1. How does changing the height of the cone affect its volume?
  2. How is the volume of a cone different from the volume of a cylinder with the same base and height?
  3. What is the significance of the factor 13\frac{1}{3} in the volume formula of a cone?
  4. How would you calculate the volume if the radius was given as a fraction?
  5. How can you derive the formula for the volume of a cone from first principles?

Tip: Always remember to check if the answer requires exact form or a rounded decimal approximation.

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Math Problem Analysis

Mathematical Concepts

Geometry
Volume of Solids
Cones

Formulas

Volume of a cone V = (1/3)πr²h

Theorems

Volume formula for cones

Suitable Grade Level

Grades 7-9