The exact formula for the volume of a certain cone, where the sum of its radius (r) and height (h) is a constant, is given by V = ⅓πr² (10-r).
This formula describes how the volume of such a cone varies based on its radius, under the specific condition that the sum of its radius and height remains constant.
Understanding the Cone's Volume Formula
The standard formula for the volume of any cone is V = ⅓πr²h, where:
- V represents the volume.
- π (pi) is a mathematical constant approximately equal to 3.14159.
- r is the radius of the cone's base.
- h is the height of the cone.
In the specific case described, the sum of the radius (r) and height (h) is constant. Let's denote this constant as C. So, r + h = C.
From this relationship, we can express the height in terms of the radius and the constant: h = C - r.
By substituting this expression for h back into the standard volume formula, we get:
V = ⅓πr²(C - r)
Comparing this general form to the provided formula V = ⅓πr²(10 - r), it implies that the constant sum (C) for this particular cone is 10 units. This means that for this specific cone, the radius and height always add up to 10 (i.e., r + h = 10).
Key Components of the Formula
The formula V = ⅓πr² (10-r) allows you to calculate the volume of such a cone if you know its radius.
- ⅓π: These are constant factors.
- r²: Represents the square of the cone's radius. As the radius increases, this term grows quadratically.
- (10-r): This term represents the effective height of the cone (since h = 10 - r). It shows that as the radius increases, the height decreases proportionally, maintaining the constant sum of 10.
Practical Implications and Examples
This formula is crucial for understanding how to optimize the volume of a cone when there's a constraint on the sum of its radius and height. For instance, if you were designing a cone-shaped container with a fixed perimeter for its cross-section (representing r+h), this formula would help you determine the optimal radius for maximum volume.
- Example Calculation:
- If the radius (r) is 3 units, the volume would be:
V = ⅓π(3)²(10-3)
V = ⅓π(9)(7)
V = 21π cubic units (approximately 65.97 cubic units) - If the radius (r) is 5 units, the volume would be:
V = ⅓π(5)²(10-5)
V = ⅓π(25)(5)
V = 125/3 π cubic units (approximately 130.90 cubic units)
- If the radius (r) is 3 units, the volume would be:
Notice how the volume changes with the radius. There is an optimal radius that yields the maximum possible volume for this cone, which can be found using calculus (by finding the derivative of V with respect to r and setting it to zero). For this formula, the maximum volume occurs when r = 20/3 or approximately 6.67 units.
Rate of Change in a Dynamic Context
While the question specifically asks for the volume formula, it is important to note that in real-world applications, geometric properties often change over time. The rate of change of the radius of this cone with respect to time (dr/dt) is stated to be 6.14. This information would be relevant if one needed to calculate the rate of change of the cone's volume over time (dV/dt).
To find dV/dt, one would use the chain rule:
*dV/dt = (dV/dr) (dr/dt)**
First, differentiate the volume formula with respect to r:
V = ⅓π(10r² - r³)
dV/dr = ⅓π(20r - 3r²)
Then, substitute this into the chain rule along with the given dr/dt = 6.14. This would allow for the calculation of how quickly the cone's volume is changing at a specific radius.
This demonstrates how static formulas for volume can be extended into dynamic scenarios, providing insights into changing systems.
