The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation S = πr√(r² + h²). 
a. How is dS/dt related to dr/dt if h is constant?

Verified step by step guidance

1. To find how dS/dt is related to dr/dt when h is constant, we start by differentiating the given equation for the lateral surface area S with respect to time t.
2. The equation for the lateral surface area is S = πr√(r² + h²). Since h is constant, we treat it as a constant during differentiation.
3. Apply the chain rule to differentiate S with respect to t: dS/dt = d/dt [πr√(r² + h²)].
4. Differentiate the expression: dS/dt = π * (d/dt [r√(r² + h²)]). Use the product rule here, as the expression is a product of πr and √(r² + h²).
5. The derivative of r√(r² + h²) with respect to t is: (dr/dt)√(r² + h²) + (r * (1/2)(r² + h²)^(-1/2) * 2r * dr/dt). Simplify this expression to find the relationship between dS/dt and dr/dt.

Key Concepts

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this context, we are interested in how the lateral surface area S of a cone changes with respect to the radius r while keeping the height h constant. This requires applying the chain rule to differentiate the equation with respect to time.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. When dealing with related rates, the chain rule allows us to express the derivative of a function in terms of the derivatives of its variables. For the equation S = πr√(r² + h²), we will differentiate S with respect to time t, leading to a relationship between dS/dt and dr/dt.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated. In this problem, we will treat S as a function of r and h, and since h is constant, we can differentiate S implicitly with respect to t. This will help us find the relationship between the rates of change of S and r.