Geometry of Circles and Spheres

This video tutorial covers related rate problems involving circles and spheres, including the inflated balloon and melting snowball problems. It explains how to calculate the rate of change of a circle's circumference and area, the rate of change of a balloon's radius as air is pumped in, and the rate of change of a snowball's diameter as it melts. The tutorial provides step-by-step solutions and emphasizes the importance of understanding the relationships between different variables in these problems.

If the radius of a circle is increasing at 5 cm/min, how fast is the circumference changing when the radius is 10 cm?

• 5π cm/min

• 10π cm/min

• 20π cm/min

• 15π cm/min

What is the formula for the circumference of a circle?

• πr²

• 2πr

• 4/3πr³

• πd

When the radius of a circle is 8 cm and increasing at 5 cm/min, how fast is the area changing?

• 100π cm²/min

• 80π cm²/min

• 40π cm²/min

• 60π cm²/min

What is the formula for the area of a circle?

• πd

• 4/3πr³

• 2πr

• πr²

Air is pumped into a spherical balloon at 450 cm³/min. How fast is the radius changing when the radius is 10 cm?

• 10/9π cm/min

• 9/10π cm/min

• 8/9π cm/min

• 9/8π cm/min

What is the formula for the volume of a sphere?

• πr²

• 2πr

• 4/3πr³

• πd

The surface area of a melting snowball is decreasing at 2 cm²/min. How fast is the diameter changing when the radius is 5 cm?

• -1/10π cm/min

• -4/10π cm/min

• -2/10π cm/min

• -3/10π cm/min

What is the formula for the surface area of a sphere?

• 4/3πr³

• πr²

• 2πr

• 4πr²

If the diameter of a sphere is twice the radius, what is the diameter when the radius is 5 cm?

• 5 cm

• 10 cm

• 15 cm

• 20 cm

What is the relationship between the diameter and radius of a sphere?

• Diameter is twice the radius

• Diameter is half the radius

• Diameter is three times the radius

• Diameter is equal to the radius