Average Rate of Change

Examples, solutions, and videos to help Grade 8 students learn how to compute the average rate of change in the height of water level when water is poured into a conical container at a constant rate.

New York State Common Core Math Grade 8, Module 7, Lesson 22

Worksheets for Grade 8

Lesson 22 Student Outcomes

• Students know how to compute the average rate of change in the height of water level when water is poured into a conical container at a constant rate.

Lesson 22 Summary

• We know intuitively that the narrower part of a cone will fill up faster than the wider part of a cone.
• By comparing the time it takes for a cone to be filled to a certain water level, we can determine that the rate of filling the cone is not constant.

Lesson 22 Classwork

Watch this video of a cone being filled with water.

• Show Video

Exercise
The height of a container in the shape of a circular cone is 7.5 ft. and the radius of its base is 3 ft., as shown. What is the total volume of the cone?
• If we knew the rate at which the cone was being filled with water, how could we use that information to determine how long it would take to fill the cone?
• Water flows into the container (in its inverted position) at a constant rate of ft3 when will the container be filled?
• Now we want to show that even though the water filling the cone flows at a constant rate, the rate of change of the volume in the cone is not constant. For example, if we wanted to know how many minutes it would take for the level in the cone to reach 1 ft., then we would have to first determine the volume of the cone when the height is 1 ft. Do we have enough information to do that?
• What equation can we use to determine the radius when the height is 1 ft.? Explain how your equation represents the situation.
• Use your equation to determine the radius of the cone when the height is 1 ft.
• Now determine the volume of the cone when the height is 1 ft.
• Calculate the number of minutes it would take to fill the cone at 1 ft. intervals. Organize your data in the table below.
• We know that the sand (rice, water, etc.) being poured into the cone is poured at a constant rate, but is the level of the substance in the cone rising at a constant rate? Provide evidence to support your answer.

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