If several quantities are related by an equation, then differentiating both sides of that equation with respect to a variable (usually \(t\), representing time) produces a relation between the rates of change of those quantities. The known rates of change are then used in that relation to determine an unknown related rate.
Example \(\PageIndex{1}\): relrate1
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Solution
Suppose that water is being pumped into a rectangular pool at a rate of 60,000 cubic feet per minute. If the pool is 300 ft long, 100 ft wide, and 10 ft deep, how fast is the height of the water inside the pool changing?
Solution: Let \(V\) be the volume of the water in the pool. Since the volume of a rectangular solid is the product of the length, width, and height of the solid, then
\[V ~=~ (300)(100)h ~=~ 30000h ~~\text{ft}^3 \nonumber \]
where \(h\) is the height of the water, as in the picture on the right. Both \(V\) and \(h\) are functions of time \(t\) (measured in minutes), and \(\dVdt = 60000~\text{ft}^3\)/min was given. The goal is to find \(\frac{d\!h}{\dt}\). Since
\[\dVdt ~=~ \ddt\,(30000h) ~=~ 30000\,\frac{d\!h}{\dt} \nonumber \]
then
\[\frac{d\!h}{\dt} ~=~ \frac{1}{30000} \dVdt ~=~ \frac{1}{30000}\cdot 60000 ~=~ 2~\text{ft/min} . \nonumber \]
Example \(\PageIndex{1}\): relrate2
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Solution
Suppose that the angle of inclination from the top of a 100 ft pole to the sun is decreasing at a rate of \(0.05\) radians per minute. How fast is the length of the pole’s shadow on the ground increasing when the angle of inclination is \(\pi/6\) radians? You may assume that the pole is perpendicular to the ground.
Solution: Let \(\theta\) be the angle of inclination and let \(x\) be the length of the shadow, as in the picture on the right. Both \(\theta\) and \(x\) are functions of time \(t\) (measured in minutes), and \(\frac{d\negmedspace\theta}{\dt} = -0.05\) rad/min was given (the derivative is negative since \(\theta\) is decreasing). The goal is to find \(\dxdt\) when \(\theta = \pi/6\), denoted by \(\dxdt\Biggr|_{\theta=\pi/6}\) (the vertical bar means “evaluated at” the value of the subscript to the right of the bar). Since
\[x ~=~ 100\;\cot\,\theta \quad\Rightarrow\quad \dxdt ~=~ -100\;\csc^2\theta \cdot \frac{d\negmedspace\theta}{\dt} ~=~ -100\;\csc^2\theta \cdot (-0.05) ~=~ 5\;\csc^2\theta \nonumber \]
then
\[\dxdt\Biggr|_{\theta=\pi/6} ~=~ 5\;\csc^2(\pi/6) ~=~ 5\;(2)^2 ~=~ 20~\text{ft/min} . \nonumber \]
Example \(\PageIndex{1}\): relrate3
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Solution
The radius of a right circular cylinder is decreasing at the rate of \(3\) cm/min, while the height is increasing at the rate of \(2\) cm/min. Find the rate of change of the volume of the cylinder when the radius is \(8\) cm and the height is \(6\) cm.
Solution: Let \(r\), \(h\), and \(V\) be the radius, height, and volume, respectively, of the cylinder. Then \(V = \pi\,r^2 \,h\). Since \(\frac{\dr}{\dt} = -3\) cm/min and \(\frac{d\negmedspace h}{\dt} = 2\) cm/min, then by the Product Rule:
\[\dVdt ~=~ \ddt (\pi\,r^2 \,h) ~=~ \left( 2\pi \,r\;\cdot\;\drdt\right)\,h ~+~ \pi\,r^2 \;\cdot\;\frac{d\negmedspace h}{\dt} \quad\Rightarrow\quad \dVdt\Biggr|_{\text{\scriptsize{$\begin{matrix}r=8\\h=6\end{matrix}$}}} ~=~ 2\pi\,(8)\,(-3)\,(6) ~+~ \pi\,(8^2)\,(2) ~=~ -160\pi~\frac{\text{\scriptsize cm}^3}{\text{\scriptsize min}} \nonumber \]
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- A stone is dropped into still water. If the radius of the circular outer ripple increases at the rate of \(4\) ft/s, how fast is the area of the circle of disturbed water increasing when the radius is \(10\) ft?
- The radius of a sphere decreases at a rate of 3 mm/hr. Determine how fast the volume and surface area of the sphere are changing when the radius is 5 mm.
- A kite \(80\) ft above level ground moves horizontally at a rate of \(4\) ft/s away from the person flying it. How fast is the string being released at the instant when \(100\) ft of string have been released?
- A \(10\)-ft ladder is leaning against a wall on level ground. If the bottom of the ladder is dragged away from the wall at the rate of \(5\) ft/s, how fast will the top of the ladder descend at the instant when it is \(8\) ft from the ground?
- A person \(6\) ft tall is walking at a rate of \(6\) ft/s away from a light which is \(15\) ft above the ground. At what rate is the end of the person’s shadow moving along the ground away from the light?
- An object moves along the curve \(y = x^3\) in the \(xy\)-plane. At what points on the curve are the \(x\) and \(y\) coordinates of the object changing at the same rate?
- The radius of a right circular cone is decreasing at the rate of \(4\) cm/min, while the height is increasing at the rate of \(3\) cm/min. Find the rate of change of the volume of the cone when the radius is \(6\) cm and the height is \(7\) cm.
- Two boats leave the same dock at the same time, one goes north at 25 mph and the other goes east at 30 mph. How fast is the distance between the boats changing when they are 100 miles apart?
- Repeat Exercise 8 with the angle between the boats being \(110\Degrees\). [[1.]]
- An angle \(\theta\) changes with time. For what values of \(\theta\) do \(\sin\,\theta\) and \(\tan\,\theta\) change at the same rate?
- Repeat Example
Example \(\PageIndex{1}\): relrate2
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Solution
- An upright cylindrical tank full of water is tipped over at a constant angular speed. Assume that the height of the tank is at least twice its radius. Show that at the instant the tank has been tipped \(45\Degrees\), water is leaving the tank twice as fast as it did at the instant the tank was first tipped. (Hint: Think of how the water looks inside the tank as it is being tipped.
