Flashcards in 7.4.1 The Pebble Problem Deck (8):
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.
- 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?