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Chapter 3: Applications of Differentiation
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Section 3.6: Related Rates
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Essentials


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Relatedrates problems are exercises designed to illustrate the use of the chain rule. Varying quantities linked by some quantitative relationship give rise to "related rates of change." The calculus content of the typical such problem is minimal; the challenge in a relatedrates problem, which is typically a "word problem," is expressing the relationship in terms of the appropriate variables.

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For example, suppose the quantity $y$ depends on the quantity $x$, and that $x$ in turn changes in time, $t$. Of course, that induces a related change in $y$, and thus, this change in $y$ is called a "related rate." In this simply stated example, $y\=y\left(x\left(t\right)\right)$, and the rate of change of $y$, namely, $\frac{\mathrm{dy}}{\mathrm{dt}}$, is given by the chain rule as $\frac{\mathrm{dy}}{\mathrm{dx}}\cdot \frac{\mathrm{dx}}{\mathrm{dt}}$. That's all the calculus involved in the typical relatedrate problem. Again, the challenge is extracting the formula $y\left(x\right)$ from the words of the statement of the problem.${}$

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One last tip: The typical relatedrate problem asks for a "related rate" at some specific moment. Almost never is this magic moment given, or need it be found. The time at which the measurement to be made is coincident with the state of one of the variables in the problem, and it is the value of that variable that is used to fix the value of the related rate. See the Examples 3.6.15 for illustrations of this "feature" of most such problems.${}$

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Examples


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Example 3.6.1

At 1:00 PM a ship sets sail due north at a speed of 14 knots, and an hour later a second ship sets sail due east at a speed of 19 knots. At what rate is the distance between these two ships increasing at 7:00 PM?

Example 3.6.2

At 1:00 PM a ship traveling at 9 knots sets sail northeast along a line that makes a $30\xb0$ angle with a line running due east. An hour later, a second ship sets sail due north, and at 11:00 PM, the distance between the ships is observed to be increasing at a rate of $97\/7$ knots. Assuming it travels at constant speed, how fast is the northbound ship traveling?

Example 3.6.3

Sand pouring from a hopper at a steady rate forms a conical pile whose height is observed to remain twice the radius of the base of the cone. When the height of the pile is observed to be 20 ft, the radius of the base of the pile appears to be increasing at the rate of a foot every two minutes. How fast is the sand pouring from the hopper?

Example 3.6.4

Helium is pumped into a spherical balloon at the constant rate of 25 cu ft per min.
At what rate is the surface area of the balloon increasing at the moment when its radius is 8 ft?

Example 3.6.5

A rightcircular conical tank, whose crosssection through its axis is shown in Figure 3.6.5(a), is being filled with water at the constant rate $\mathrm{\λ}$.
At time $\stackrel{\^}{t}$, find $\stackrel{\.}{h}\left(\stackrel{\^}{t}\right)$, the rate of change of the height of the water, where $\stackrel{\^}{t}$ is the moment when the volume is $k$ times the volume of the tank, $0<k\le 1$.


Figure 3.6.5(a) Conical tank




The dimensions of the tank and the rate of fill are all in consistent units. The height of the tank is $H$, while the radius of the opening is $R$. The varying radius of the circle at the level of the water is $r\left(t\right)$ (green dotted line in Figure 3.6.5(a)), and the varying height of the water is $h\left(t\right)$.
Hint: The volume of the tank is $\frac{\mathrm{\π}}{3}{R}^{2}H$






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