CALCULUS

RELATED RATES MATCH ACTIVITY

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Related Rates Sort & Match ACTIVITY

Use this activity as review for Related Rates. The concepts will help your students work through the rules for setting up problems, implicit differentiation with respect to time, (including Chain Rule), and solving the basic types of related rates problems.

Teaching Suggestions:• Use this activity as a group quiz activity for 2-4 per group• Use the cards as a review activity prior to assessing students

You may be interested in purchasing the Applications of Derivatives Scavenger Hunt Activity

Related Rates By Jean Adams

This Sort & Match Activity is designed for AP Calculus AB students. There are eight task cards to present a typical related rate situation. Students will match the task card with the question, to an equation task card, and its related derivative card. When students are finished with this portion, have them complete the activity by listing the unknown in each question, stating the givens, then solving the problem for each situation.

Directions:

1) Print and laminate the cards on colored card stock.2) Then, copy a recording sheet for each student, or group of students.3) The activity is intended as a review for Related Rates.

Related Rates Key

Related Rate Problem General Equation Derivative

R1 E5 D8

R2 E6 D4

R3 E2 D5

R4 E3 D7

R5 E7 D2

R6 E1 D6

R7 E8 D1

R8 E4 D3

© 2014 Flamingo Math.com Jean Adams

R1

Assume the radius 𝑟𝑟 of a sphere is expanding at a rate of 12 in/min. Find the rate at which the volume is changing with respect to time when 𝑟𝑟 = 5 𝑖𝑖𝑖𝑖.

R2

Water pours into a conical tank of height 10 m and radius 4 m at a rate of 6m3/min. Find the rate the water level is rising, when the level is 5 m high.

R3

Water pours into a fish tank at a rate of 0.3 m3/min. How fast is the water level rising if the base of the tank is a rectangle of dimensions 3 ×4 meters?

R4

A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off position. At a certain moment, the angle between the observer’s line of sight and the horizontal is 𝜋𝜋

5 and is changing at a rate of

0.2 radians/min. How fast is the balloon rising at this moment?

© 2014 Flamingo Math.com Jean Adams

R5

Kevin and Austin are trying to get into an upstairs window. They lean a 15 ft. ladder against the wall. But, the ladder is slipping so that its base moves away from the wall at a rate of 3 ft/sec. How fast is the angle formed by the ladder and the ground decreasing when the ladder’s base is 9 ft. from the wall?

R6

Sonia and Sana are in speedboats located at the center of Lake Down. At time 𝑡𝑡 = 0 minutes, Sonia begins to travel north at a speed of 30 mph and Sana travels east at a speed of 26 mph. At what rate is the distance between them increasing when 𝑡𝑡 = 12 minutes?

R7

The height of a cylinder with a radius of 6 cm is increasing at a rate of 2 cm/min. Find the rate of change of the volume with respect to time when the height is 14 centimeters.

R8

Oil spilled from a tanker is spreading in a circle whose area increases at a constant rate of 8 square miles per hour. How fast is the radius of the spill increasing when the area is 16 square miles?

© 2014 Flamingo Math.com Jean Adams

E1

𝑥𝑥2 + 𝑦𝑦2 = 𝑧𝑧2

E2

𝑉𝑉 = 𝐵𝐵ℎ

E3

𝑥𝑥 tan𝜃𝜃 = 𝑦𝑦

E4

𝐴𝐴 = 𝜋𝜋𝑟𝑟2

E5

𝑉𝑉 =43𝜋𝜋𝑟𝑟3

E6

𝑉𝑉 =13𝜋𝜋𝑟𝑟2ℎ

E7

cos𝜃𝜃 =𝑥𝑥𝑟𝑟

E8

𝑉𝑉 = 𝜋𝜋𝑟𝑟2ℎ

© 2014 Flamingo Math.com Jean Adams

D1

𝑑𝑑𝑉𝑉𝑑𝑑𝑡𝑡

= 𝜋𝜋𝑟𝑟2𝑑𝑑ℎ𝑑𝑑𝑡𝑡

D2

−𝑟𝑟 sin𝜃𝜃𝑑𝑑𝜃𝜃𝑑𝑑𝑡𝑡

=𝑑𝑑𝑥𝑥𝑑𝑑𝑡𝑡

D3

𝑑𝑑𝐴𝐴𝑑𝑑𝑡𝑡

= 2𝜋𝜋𝑟𝑟𝑑𝑑𝑟𝑟𝑑𝑑𝑡𝑡

D4

𝑑𝑑𝑉𝑉𝑑𝑑𝑡𝑡

=13𝜋𝜋 �2𝑟𝑟ℎ

𝑑𝑑𝑟𝑟𝑑𝑑𝑡𝑡

+ 𝑟𝑟2𝑑𝑑ℎ𝑑𝑑𝑡𝑡�

D5

𝑑𝑑𝑉𝑉𝑑𝑑𝑡𝑡

=𝑑𝑑𝐵𝐵𝑑𝑑𝑡𝑡

ℎ + 𝐵𝐵𝑑𝑑ℎ𝑑𝑑𝑡𝑡

D6

2𝑥𝑥𝑑𝑑𝑥𝑥𝑑𝑑𝑡𝑡

+ 2𝑦𝑦𝑑𝑑𝑦𝑦𝑑𝑑𝑡𝑡

= 2𝑧𝑧𝑑𝑑𝑧𝑧𝑑𝑑𝑡𝑡

D7

tan𝜃𝜃𝑑𝑑𝑥𝑥𝑑𝑑𝑡𝑡

+ 𝑥𝑥 sec2 𝜃𝜃𝑑𝑑𝜃𝜃𝑑𝑑𝑡𝑡

=𝑑𝑑𝑦𝑦𝑑𝑑𝑡𝑡

D8

𝑑𝑑𝑉𝑉𝑑𝑑𝑡𝑡

= 4𝜋𝜋𝑟𝑟2𝑑𝑑𝑟𝑟𝑑𝑑𝑡𝑡

© 2014 Flamingo Math.com Jean Adams

Related Rates Lab Activity Name _____________________________________________

Date ____________________________ Period __________

Part 1: Complete the table to record your matches.

Related Rate Problem

General Equation

Derivative

R1

R2

R3

R4

R5

R6

R7

R8

Part 2: For each of the 8 problems given:

A.) Determine the unknown in the question.

B.) List the givens. (Hint: If a value is a constant, then the rate of change is 0.)

C.) Solve the problem for each situation.

© 2014 Flamingo Math.com Jean Adams

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