Section 3.10 Related Rates
Example 3.10.1.
A spherical balloon is being inflated. The volume of the balloon is \(V=\frac{4}{3}\pi r^3\text{.}\)
What is the rate of change of the volume at a time when the radius is 4 inches and is increasing at a rate of \(\frac{1}{3}\) inches per minute?
What is the rate of change of the radius at a time when the radius is 3 inches if the volume is increasing at a rate of 10 cubic inches per minute.
Example 3.10.2.
Icing is being squeezed out of a conical tube (\(V = \frac{1}{3}\pi r^2h\)) at a rate of \(3~\text{cm}^3/\text{sec}\text{.}\) Initially, the icing fills the tube to a depth of 23 cm with a radius of 12 cm at the widest point. How quickly is the height of icing in the tube changing when it is 15 cm deep?
Example 3.10.3. Number 42.
A highway patrol plane flies 3 mi above a level, straight road at a steady 120 mi/h. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from the plane to the car is 5 mi, the line-of-sight distance is decreasing at the rate of 160 mi/h. Find the car's speed along the highway.
Example 3.10.4. Number 29.
Water is flowing at the rate of \(6 \text{m}^3/\text{min}\) from a reservoir shaved like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following questions given that the water in a hemispherical bowl of radius \(R\) is \(\displaystyle V = \frac{\pi}{3} y^2 (3R-y)\) when the water is \(y\) meters deep.
At what rate is the water level changing when the water is 8 m deep?
What is the radius \(r\) of the water's surface when the water is \(y\) m deep?
At what rate is the radius \(r\) changing when the water is 8 m deep?
Example 3.10.6. Number 33.
A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance \(s(t)\) between the bicycle and balloon increasing 3 sec later?