Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

Previous problemNext problem

2:49 minutes

Problem 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 guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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.

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.

Watch next

Master Intro To Related Rates with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

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.

Textbook Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.

b. Interpret the meaning of your results in part (a).

Textbook Question

Explain Rolle’s Theorem with a sketch.

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

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

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

<IMAGE>

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

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?

<IMAGE>

Textbook Question

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

<IMAGE>

Show more practices