Rate of Change in Geometry

The video tutorial covers two examples of related rates problems. The first example involves a baseball diamond, where a player runs from first to second base, and the task is to find the rate at which the distance from home plate changes. The solution involves using the Pythagorean theorem and calculus derivatives. The second example deals with a sphere whose radius increases at a constant rate, and the goal is to determine the rate of volume increase. The tutorial explains the use of derivatives and algebraic manipulation to solve these problems.

What shape is the baseball diamond described in Example 5?

• Circle

• Triangle

• Square

• Rectangle

How long are the sides of the baseball diamond?

• 60 feet

• 90 feet

• 120 feet

• 150 feet

At what speed is the player running from first to second base?

• 30 feet per second

• 20 feet per second

• 25 feet per second

• 28 feet per second

Which theorem is used to find the rate of change of distance in Example 5?

• Fundamental theorem of calculus

• Mean value theorem

• Binomial theorem

• Pythagorean theorem

What is the value of x when the player is 30 feet from second base?

• 30

• 60

• 120

• 90

What is the rate of change of the distance from home plate when the player is 30 feet from second base?

• 30 root 13 feet per second

• 90 feet per second

• 28 feet per second

• 56 root 13/13 feet per second

In Example 6, at what rate is the radius of the sphere increasing?

• 0.5 cm/s

• 0.4 cm/s

• 0.3 cm/s

• 0.2 cm/s

What is the radius of the sphere when calculating the rate of volume increase?

• 20 cm

• 5 cm

• 15 cm

• 10 cm

What is the formula for the rate of change of volume of a sphere?

• 3 π r^2

• 4 π r^2 dr/dt

• 2 π r

• 4/3 π r^3

What is the rate of change of volume when the radius is 10 cm?

• 12 π cm³/s

• 14 π cm³/s

• 16 π cm³/s

• 18 π cm³/s