Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
b. Evaluate this derivative when a=6 and b=10.

Verified step by step guidance

1. First, identify the formula for the volume of the torus: \( V = \frac{\pi^2 (b+a)(b-a)^2}{4} \).
2. To find the derivative of the volume with respect to the outer radius \( b \), apply the product rule and chain rule to the expression \( V = \frac{\pi^2}{4} (b+a)(b-a)^2 \).
3. Differentiate \( (b+a) \) with respect to \( b \), which is simply 1, since \( a \) is a constant.
4. Differentiate \( (b-a)^2 \) with respect to \( b \) using the chain rule: \( 2(b-a) \cdot 1 \).
5. Substitute \( a = 6 \) and \( b = 10 \) into the derivative expression to evaluate it at these values.

Key Concepts

Volume of a Torus

The volume of a torus is calculated using the formula V=π²(b+a)(b−a)²/4, where 'a' is the inner radius and 'b' is the outer radius. This formula derives from integrating the area of circular cross-sections of the torus, taking into account the geometry of the shape. Understanding this formula is essential for evaluating the volume based on given radii.

Derivative

A derivative represents the rate of change of a function with respect to a variable. In this context, evaluating the derivative of the volume function with respect to the radii 'a' and 'b' will provide insights into how changes in these dimensions affect the volume of the torus. Mastery of differentiation techniques is crucial for solving problems involving rates of change.

Evaluation of Functions

Evaluating a function involves substituting specific values into the function to compute a result. In this case, substituting a=6 and b=10 into the volume formula allows for the calculation of the torus's volume. This concept is fundamental in calculus, as it connects theoretical functions to practical numerical results.