Circle Equations and Radius Calculations

In this video, Patrick explains how to solve a problem involving circular ripples created by a stone dropped in water. The problem is divided into two scenarios: one where the radius of the circle grows at a constant rate, and another where the area of the circle grows at a constant rate. Patrick demonstrates how to find the equation of the circle in both scenarios, using the formula x^2 + y^2 = r^2. He explains the steps to calculate the radius and area growth, and how to apply these to find the circle's equation after a given time.

What shape do the waves form when a stone is dropped into water?

• Square

• Triangle

• Circle

• Rectangle

What is the format of the circle's equation used in the video?

• x + y = r

• x^2 + y^2 = r^2

• x^2 - y^2 = r^2

• x^2 + y = r^2

In the first scenario, how fast does the radius grow?

• 2 cm/s

• 1 cm/s

• 4 cm/s

• 3 cm/s

After how many seconds is the radius calculated in the first scenario?

• 5 seconds

• 4 seconds

• 6 seconds

• 3 seconds

What is the radius of the circle after 6 seconds in the first scenario?

• 16 cm

• 14 cm

• 12 cm

• 10 cm

What is the equation of the circle after 6 seconds in the first scenario?

• x^2 + y^2 = 196

• x^2 + y^2 = 100

• x^2 + y^2 = 144

• x^2 + y^2 = 256

In the second scenario, how much does the area grow per second?

• 10π cm²

• 20π cm²

• 15π cm²

• 5π cm²

How many seconds are considered in the second scenario for area growth?

• 2 seconds

• 5 seconds

• 3 seconds

• 4 seconds

What is the total area after 3 seconds in the second scenario?

• 35π cm²

• 30π cm²

• 25π cm²

• 20π cm²

What is the equation of the circle after 3 seconds in the second scenario?

• x^2 + y^2 = 35

• x^2 + y^2 = 30

• x^2 + y^2 = 25

• x^2 + y^2 = 20