Textbook Question
Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.
a. Determine the average and marginal costs for x = 3000 lawn mowers.
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4. Applications of Derivatives
Related Rates
2:49 minutesProblem 96a
Textbook Question
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 guidance1
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.
The equation for the lateral surface area is S = πr√(r² + h²). Since h is constant, we treat it as a constant during differentiation.
Apply the chain rule to differentiate S with respect to t: dS/dt = d/dt [πr√(r² + h²)].
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).
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.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Intro To Related Rates
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.
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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.
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