CramX
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 7.4.1 The Pebble Problem

AP Calculus AB: 7.4.1 The Pebble Problem

Mathematics8 CardsCreated 3 months ago

This content explains how to solve related rates problems by connecting the rate of change of one quantity to another using derivatives. It includes examples such as finding the rate of change of the area of a ripple with expanding radius and the area of an isosceles triangle as its included angle changes.

PrintEmbedImportReport

The Pebble Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • The three steps to problem-solving are understanding what you want, determining what you know, and finding a connection between the two.

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/8

Key Terms

Term
Definition

The Pebble Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • The three steps to problem-...

note

  • If you drop a stone into a body of water, ripples form across the surface of the water.

  • Suppose you are told that the radius...

Two identical twelve-inch rulers on a desk form an isosceles triangle with included angle θ. If Mickie moves the rulers so that θ increases at the rate of 2.5 radians per second, what is the rate of change of the area of the triangle when θ = π / 4 radians?

Note: the area of an isosceles triangle with legs of length l and included angle θ is given by the formula
A = 1/2l^2sin θ

127 square inches per second

Water is draining out of a conical tank at a constant rate of 120 cubic inches per minute. How fast is the level of water in the tank decreasing when there are 7 inches of water in the tank?

The level of water in the tank is decreasing at a rate of 3.1 inches per minute.

Farmer Bob’s square plot of land is slowly eroding away. Worried about the future of his farm, Farmer Bob measures the rate of erosion and finds that the length of each side of his square plot is decreasing at the constant rate of 2 feet / year. If he currently owns 250,000 square feet of land, what is the current rate of change of the area of Farmer Bob’s land?

Farmer Bob is losing 2,000 square feet of land per year.

A balloon is being inflated from a helium tank at a constant rate of 50 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches? Assume that the balloon is a perfect sphere.

1/2π inches per minute

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck
Mathematics

Unit 2 Progress Check: MCQ Part A

To determine the point where the tangent line to the graph of the function 𝑓f has a slope of 2 for 𝑥 >0 x>0, we use the fact that the slope of the tangent line at any point is given by the derivative 𝑓′(𝑥)f′(x).

15 cards
View Deck
Mathematics

Customary Liquid Measurement Units

This set of flashcards outlines key relationships between customary units of liquid measurement, including pints, cups, quarts, and gallons. It provides conversion facts to help understand how these units relate to one another in the U.S. measurement system.

3 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 7.4.1 The Pebble Problem
TermDefinition

The Pebble Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • The three steps to problem-solving are understanding what you want, determining what you know, and finding a connection between the two.

note

  • If you drop a stone into a body of water, ripples form across the surface of the water.

  • Suppose you are told that the radius of a ripple is increasing at a rate of 6 inches per second.

  • What is the rate of change in the area enclosed by the ripple?

  • This is an example of a related rate. Related rate questions ask you to find information about one rate given information about another.

  • You want to know the rate of change in the area enclosed by the ripple.

  • You are given the rate of change of the radius of the ripple.

  • You also know the formula for the area of a circle.

  • Since area is expressed in terms of the radius, you can take the derivative of the area equation with respect to time. Notice that the derivative of the area equation includes two variables: the radius, and the rate of change of the radius.

  • Substitute the values for the radius and the rate of change into the equation to find the rate of change in the area.

Two identical twelve-inch rulers on a desk form an isosceles triangle with included angle θ. If Mickie moves the rulers so that θ increases at the rate of 2.5 radians per second, what is the rate of change of the area of the triangle when θ = π / 4 radians?

Note: the area of an isosceles triangle with legs of length l and included angle θ is given by the formula
A = 1/2l^2sin θ

127 square inches per second

Water is draining out of a conical tank at a constant rate of 120 cubic inches per minute. How fast is the level of water in the tank decreasing when there are 7 inches of water in the tank?

The level of water in the tank is decreasing at a rate of 3.1 inches per minute.

Farmer Bob’s square plot of land is slowly eroding away. Worried about the future of his farm, Farmer Bob measures the rate of erosion and finds that the length of each side of his square plot is decreasing at the constant rate of 2 feet / year. If he currently owns 250,000 square feet of land, what is the current rate of change of the area of Farmer Bob’s land?

Farmer Bob is losing 2,000 square feet of land per year.

A balloon is being inflated from a helium tank at a constant rate of 50 cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches? Assume that the balloon is a perfect sphere.

1/2π inches per minute

Suppose that x and y are differentiable with respect to t. If x^3+y^3=9 and dy/dt=2, what is the value of dx/dt when y=1?

dx/dt=−1/2

A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 3 inches per second. By how many inches did the circumference of the ripple increase between 0 and 2 seconds?

12π in.