7.4.4 The Blimp Problem

Question 1

The Blimp Problem

Answer 1

• Related rate problems involve using a known rate of change to find an associated rate of change.
• Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.

Question 2

note

Answer 2

• A blimp traveling overhead is tied to a rope. The rope is let out at a rate of 3 feet per second.
• Assuming the blimp remains at a constant altitude of 800 feet, how fast is the blimp moving when 100 feet of rope have been let out?
• This situation can be modeled by a right triangle. Notice that it is not important to keep your diagram to scale. In fact, drawing to scale might mislead you, giving you false information.
• The Pythagorean theorem relates the lengths of the sides of a right triangle to each other.
• Use implicit differentiation to find the derivative of b with respect to time. Then substitute the known values into the equation.

Question 3

Suppose that a certain clock has a minute hand that is 4 inches long and an hour hand that is 3 inches long. What is the rate of change of the distance between the tips of the minute and hour hands at 2:00?

Answer 3

−16.6 inches / hour

Question 4

Laura is hitting a golf ball at the driving range. Suppose that the altitude of the ball is given in feet by the equation y = 200t − 40t ^2, where t is the time in seconds after she hits the ball. Suppose also that the horizontal velocity of the ball is constant and equal to 80 feet per second. At what rate is the angle of elevation of the ball changing 2 seconds after Laura hits the ball?

Answer 4

−0.15 radians / second

Question 5

Two cars leave an intersection at the same time, one headed west and the other north. The westbound car is moving at 40 mph and the northbound car is moving at 50 mph. Fifteen minutes later, what is the rate of change in the perimeter of the right triangle created using the two cars and the intersection?

Answer 5

154.02 mph

Question 6

An object is moving along the graph of the curve x^2y^2/2x−y=1. At a particular moment, the particle is at (1,1) and dx/dt=5. Find the value of dy/dt at this moment.

Answer 6

0

Question 7

A spherical balloon is being inflated from a compressor. Suppose the volume of the balloon is increasing at a constant rate of 10 cubic inches per second. At what rate is the surface area of the balloon increasing when its radius is 6 inches?

Answer 7

3.3 square inches per second