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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 96a
Chapter 3, Problem 96a

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?

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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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Key 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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Intro to the Chain Rule

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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Finding The Implicit Derivative
Related Practice
Textbook Question

Resistors connected in parallel If two resistors of R₁ and R₂ ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation


1/R = 1/R₁ + 1/R₂


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If R₁ is decreasing at the rate of 1ohm/sec and R₂ is increasing at the rate of 0.5 ohm/sec, at what rate is R changing when R₁ = 75 ohms and R₂ = 50 ohms?

Textbook Question

The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.


a. How is dS/dt related to dr/dt if h is constant?

Textbook Question

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation:


T = 2π√(L/g),


where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant,


dL/du = kL.


Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT/2.

Textbook Question

Right circular cone 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.


c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Textbook Question

If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).

Textbook Question

Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.


a. What is the relation between the variables h and r in the figure?


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