To find the radius of a cone given its height and volume, we use the formula for the volume of a cone, V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height. Rearranging the formula to solve for r, we get r = √(3v/πh).
Substituting the given values, V = 1570 cm³ and h = 15 cm, and using π = 3.14, we find r = √[(3 × 1570)/(3.14 × 15)].
Calculating this gives the radius of the base of the cone.

Let’s discuss in detail

The task of determining the radius of a cone from its volume and height is a classic problem in geometry, illustrating the practical application of mathematical formulas. A cone is a three-dimensional shape with a circular base that tapers smoothly to a point, known as the apex. The volume of a cone is a measure of the space it occupies, and it’s directly related to the cone’s height and the radius of its base. This relationship is governed by a specific mathematical formula, making it possible to calculate one dimension if the others are known.

Understanding the Volume Formula of a Cone

The volume of a cone is calculated using the formula V = (1/3)πr²h, where V is the volume, r is the radius of the base, h is the height of the cone, and π is a mathematical constant approximately equal to 3.14. This formula is derived from the principle that the volume of a cone is a third of the volume of a cylinder with the same base and height. The formula is a cornerstone in geometry, enabling the calculation of the volume for various applications.

Rearranging the Formula to Find the Radius

To find the radius of a cone when the volume and height are known, the volume formula needs to be rearranged. By isolating the radius on one side of the equation, the formula becomes r = √(3v/πh). This rearrangement allows for the calculation of the radius based on the known volume and height. It’s a straightforward process that involves substituting the known values into the formula and solving for r.

Applying the Formula to a Specific Problem
In the given problem, the height (h) of the cone is 15 cm, and the volume (V) is 1570 cm³. By substituting these values into the rearranged formula, along with the value of π as 3.14, the calculation becomes r = √[(3 × 1570)/(3.14 × 15)]. This calculation will yield the radius of the cone’s base, which is the unknown dimension we seek to find.