Module 16 - Related Rates

Module 16 - Answers

Lesson 1

Answer 1

16.1.1 The bottom of the ladder moves at a constant rate while the top of the ladder falls at an increasing rate.

Answer 2

16.1.2 The top is moving at approximately -1.061 ft/sec when the bottom is 5 ft from the wall.

Answer 3

16.1.3

Answer 4

16.1.4 Because the ladder is 15 feet long and the bottom moves at 3 ft/sec, the top of the ladder hits the ground after 5 seconds. The first derivative of the position of the top is undefined at this time. As t approaches 5 seconds, the velocity of the top grows without bound, so in theory the top is going infinitely fast when it hits the ground.

Lesson 2

Answer 1

16.2.1 The position of the top of the ladder at time t is modeled by X1T = 0, Y1T = -16T²+15.

The position of the bottom of the ladder at time t is modeled by , Y2T = 0.

Answer 2

16.2.2 The top of the ladder is falling at an increasing rate while the bottom is moving at a decreasing rate.

Answer 3

16.2.3 The bottom of the ladder is moving at approximately 21.201 ft/sec, 17.258 ft/sec, and 8.566 ft/sec, respectively.

Lesson 3

Answer 1

16.3.1 The two ships are approximately 61 nautical miles apart after two hours.

Answer 2

16.3.2 The ships are moving apart at the constant rate of about 30.5123 knots at all times.

Answer 3

16.3.3 The ships are about 67.8 nautical miles apart after 2 hours. They are moving apart at a constant rate of 30.5017 knots. This rate is slightly slower than the previous rate, which was 30.5123 knots.

Self Test

Answer 1

The position of the southbound car can be modeled by x₁ = 0, y₁ = 15 - 45t.

Answer 2

The position of the westbound car can be modeled by x₂ = 24 - 52t, y₂ = 0.

Answer 3

The car coming from the North will reach the intersection first.

Answer 4

The distance between the cars can be modeled by

Answer 5

The distance between the two cars is decreasing at a rate of 63.739 mph because the derivative of the distance function is -63.739 when t = 0.25.