We begin our work in Chapter 3 (Using the Derivative) by seeing how the derivative may be employed to relate the rates of two different quantities that are related and each changing as time varies. The main idea here is: if two quantities are related to one another, and each is changing as time changes, then the rates at which each quantity is changing must be related. Hence, we consider a class of problems known as related rates problems.
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
Use basic geometry results such as the Pythagorean Theorem, formulas for the area of familiar figures, and trigonometry to establish relationships among quantities of interest.
The blue balloon and the red balloon clearly blow up at different rates. Suppose time \(t\) is given in seconds. In one balloon, the radius is given by the equation \(r_{1}(t)=\frac{1}{5}t\) cm. In the other, the radius is given by the equation \(r_{2}(t) = \frac{4}{3}\sqrt[3]{t}\) cm. Which is which?
The volumes of each balloon are changing at different rates. In one case, \(V_{1}(t) = 10t \ cm^{3}\text{.}\) In the other, the volume is given by \(V_{2}(t) = \frac{4}{375}\pi t^{3} \ cm^{3}\text{.}\) Which is which?
If \(dV/dt\) is constant when filling the balloon (imagine a machine pumping it up at a uniform rate), which of the two balloons (red or blue) is the most reasonable model?
How fast is the area of the oil slick increasing when the radius is 25 m? Note that the dotted lines in the applet illustrate the size of the oil slick at the moment you are computing how fast the area of this slick is increasing.
A 16-ft ladder leans against a wall. Suppose that the bottom of the ladder moves (slides) away from the wall at a constant rate. Explorethis applet to simulate this situation.
Suppose \(h(t)\) represents the height, in feet, at which the ladder touches the wall at time \(t\) seconds. If \(x(t)\text{,}\) the distance in feet between the wall and the base of the ladder at time \(t\text{,}\) is increasing, is \(dh/dt\) positive or negative?
Suppose the bottom of the ladder is 5 ft from the wall at time \(t = 0\) seconds and it slides away from the wall at a constant rate of \(3 \ ft/s\text{.}\) Find the velocity of the top of the ladder at time \(t = 1\) second.
Prompt Copilot “Make up a typical related rates problem from calculus that involves a ladder sliding down a wall. Solve it by showing me step-by-step the process.” Draw and label diagrams that help you understand the solution that the AI produces.
Prompt Copilot “Make up an atypical, unique related rates problem that might be asked in a college-level calculus class. Then, solve it showing me step-by-step the process.”
Checkpoint74.📝 [Submit] Differentiate with Respect to Time.
In each related-rates problem, the first step is often to differentiate an equation relating two or more quantities. Match each equation with its derivative with respect to time.
Water pours into a conical tank at a constant rate of \(10 \ ft^{3}\) per minute. The tank is ten feet tall and, at its widest, has a radius of 4 feet. How fast is the water level rising when it is 5 feet high? Explore how water fills a conical tank with this applet. Watch this video on filling up a tank (5:47) for the solution to a similar problem.
Since \(x(t)^{2} + h(t)^{2} = 16^{2}\text{,}\) it follows that \(2x(t)\frac{dx}{dt}+ 2h(t)\frac{dh}{dt}= 0\text{.}\) If \(\frac{dx}{dt}\) is constant, then \(\frac{dh}{dt}= -\frac{x(t)}{h(t)}\frac{dx}{dt}\text{.}\) The ladder falls and as \(dx/dt\) increases, \(dh/dt\) decreases.
We know that \(dV/dt = 10\) where \(V\) represents the volume of water in the tank at time \(t\text{.}\) Since \(V = \frac{1}{3}\pi r(t)^{2} h(t)\) where \(\frac{r(t)}{h(t)}= \frac{4}{10}\) we have that
So, \(\displaystyle \frac{dV}{dt}= \frac{4}{75}\pi \cdot 3 h(t)^{2} \frac{dh}{dt}\text{.}\) We solve for \(\frac{dh}{dt}\) and evaluate for \(h=5\) feet. But
