Related Rates - Inflated Balloon & Melting Snowball Problem - Surface Area & Volume | Summary and Q&A

TL;DR

This video explains how to solve related rate problems involving circles and spheres by finding rates of change of different variables.

Key Insights

• ☠️ Differentiating equations helps find rates of change in related rate problems.
• ⭕ The circumference of a circle is directly related to its radius.
• ⭕ The area of a circle is directly related to the square of its radius.
• 🧊 The volume of a sphere is directly related to the cube of its radius.
• ❎ The surface area of a sphere is directly related to the square of its diameter.
• 📏 Differentiating involves applying the chain rule and simplifying equations.
• ☠️ Negative rates of change indicate a decrease, while positive rates of change indicate an increase.

Transcript

in this video we're going to go over related rate problems dealing with circles spheres the inflated balloon and the melting snowball problem so let's start with this one the radius of a circle is increasing at five centimeters per minute how fast is the circumference of the circle changing when the radius is ten so let's write down what we know so... Read More

Questions & Answers

Q: What is the equation for the circumference of a circle?

The equation for the circumference of a circle is c = 2πr, where c is the circumference and r is the radius.

Q: How do you find the rate at which the circumference is changing?

To find the rate at which the circumference is changing, differentiate the circumference equation with respect to time and substitute the given values.

Q: How do you find the rate at which the area of a circle is changing?

The equation for the area of a circle is A = πr². Differentiate the area equation with respect to time and substitute the given values to find the rate of change.

Q: How do you find the rate at which the radius of a balloon is changing?

The equation for the volume of a sphere is V = (4/3)πr³. Differentiate the volume equation with respect to time and substitute the given values to find the rate at which the radius is changing.

Q: How do you find the rate at which the diameter is changing?

To find the rate at which the diameter is changing, differentiate the equation relating the surface area of a sphere to the diameter with respect to time and substitute the given values.

Summary & Key Takeaways

• Part A: The radius of a circle is increasing at 5 cm/min. Find the rate at which the circumference is changing when the radius is 10 cm.

• Part B: The radius of a circle is increasing at 5 cm/min. Find the rate at which the area is changing when the radius is 8 cm.

• Part C: Air is pumped into a spherical balloon at a rate of 450 cubic cm/min. Find the rate at which the radius of the balloon is changing when the radius is 10 cm.

• Part D: The surface area of a melting snowball is decreasing at 2 cm²/min. Find the rate at which the diameter is changing when the radius is 5 cm.