The following problems involve the concept of Related Rates. In short, Related Rates problems combine word problems together with Implicit Differentiation, an application of the Chain Rule. Recall that if $ y=f(x) $, then $ D \{y \} = \displaystyle{ dy \over dx } = f'(x)=y' $. For example, implicitly differentiating the equation
$$ y^3+y^2= y+1$$
would be
$$ D\{y^3+y^2\} = D\{ y+1 \} \ \ \ \ \longrightarrow $$
$$ 3y^2 \cdot y'+ 2y \cdot y' = y' + 0 $$
If $ x=f(t) $ and $ y=g(t) $, then $ D\{x\} = \displaystyle{ dx \over dt } = f'(t) $ and $ D\{y\} = \displaystyle{ dy \over dt } = g'(t) $ . For example, implicitly differentiating the equation
$$ x^3+y^2= x+y+3$$
would be
$$ D\{x^3+y^2\} = D\{ x+y+3 \} \ \ \ \ \longrightarrow $$
$$ 3x^2 \cdot \displaystyle{ dx \over dt } + 2y \cdot \displaystyle{ dy \over dt } = \displaystyle{ dx \over dt } + \displaystyle{ dy \over dt } + 0 $$

In all of the following Related Rates Problems, it will be assumed that each variable function $y$ is a function of time $t$. For that reason, I will always use Leibniz notation and not the ambiguous prime notation for derivatives, i.e., i will use
$$ \displaystyle{ dy \over dt } \ \ \ \ instead \ of \ \ \ \ y' $$
Here is my strategy for approaching and solving Related Rates Problems:

- 1.) Read the problem slowly and carefully.

- 2.) Draw an appropriate sketch.

- 3.) Introduce and define appropriate variables. Use variables if quantities are changing. Use constants if quantities are not changing.

- 4.) Read the problem again.

- 5.) Clearly label the sketch using your variables.

- 6.) State what information is given in the problem.

- 7.) State what information is to be determined or found.

- 8.) Use a given equation or create an appropriate equation relating the given variables.

- 9.) Differentiate this equation with respect to the time variable $t$.

- 10.) Plug in the given rates and numbers to the differentiated equation.

- 11.) Solve for the unknown rate.

- 12.) Put proper units on your final answer.

EXAMPLE 1: Consider a right triangle which is changing shape in the following way. The horizontal leg is increasing at the rate of $ 5 \ in./min. $ and the vertical leg is decreasing at the rate of $ 6 \ in./min $. At what rate is the hypotenuse changing when the horizontal leg is $ 12 \ in. $ and the vertical leg is $ 9 \ in. $ ?

Draw a right triangle with legs labeled $x$ and $y$ and hypotenuse labeled $z$, and assume each edge is a function of time $t$.

GIVEN: $ \ \ \ \displaystyle{ dx \over dt } = 5 \ in./min. \ $ and $ \ \displaystyle{ dy \over dt }= -6 \ in./min. $

FIND: $ \ \ \ \displaystyle{ dz \over dt } $ when $ x=12 \ in. $ and $ y=9 \ in $.

Use the Pythagorean Theorem to get the equation $$ x^2 + y^2 = z^2 $$

Now differentiate this equation with repect to time $t $ getting $$ D \{ x^2 + y^2\} = D \{z^2\} \ \ \ \longrightarrow $$ $$ 2x \displaystyle{ dx \over dt } + 2y \displaystyle{ dy \over dt } = 2z \displaystyle{ dz \over dt } \ \ \ \longrightarrow \ \ $$ (Multiply both sides of the equation by $1/2$.) $$ x \displaystyle{ dx \over dt } + y \displaystyle{ dy \over dt } = z \displaystyle{ dz \over dt } \ \ \ \ \ \ \ \ \ \ \ \ \ \ (DE) $$ Now let $ x=12 $ and $ y=9 $ and solve for $z$ using the Pythagorean Theorem.

$$ 12^2+9^2= z^2 \ \ \ \longrightarrow \ \ \ z^2=225 \ \ \ \longrightarrow \ \ \ z=15 $$ Plug in all given rates and values to the equation $(DE)$ getting $$ (12)(5) + (9)(-6) = (15) \displaystyle{ dz \over dt } \ \ \ \longrightarrow $$ $$ 6 = 15 \displaystyle{ dz \over dt } \ \ \ \longrightarrow $$ $$ \displaystyle{ dz \over dt } = {6 \over 15} = { 2 \over 5} \ in/min. $$

In the list of Related Rates Problems which follows, most problems are average and a few are somewhat challenging.

- PROBLEM 1 : The edge of a square is increasing at the rate of $ \ 3 \ cm/sec $. At what rate is the square's

$ \ \ \ \ $ a.) perimeter changing

$ \ \ \ \ $ b.) area changing

when the edge of the square is $10 \ cm.$ ?Click HERE to see a detailed solution to problem 1.

- PROBLEM 2 : The length of a rectangle is increasing at the rate of $ \ 4 \ ft/hr. $ and the width of the rectangle is decreasing at the rate of $ \ 3 \ ft/hr. $ At what rate is the rectangle's

$ \ \ \ \ $ a.) perimeter changing

$ \ \ \ \ $ b.) area changing

when the length is $ \ 8 \ ft.$ and the width is $ \ 5 \ ft.$ ?Click HERE to see a detailed solution to problem 2.

- PROBLEM 3 : Leg one of a right triangle is decreasing at the rate of $ \ 5 \ in/sec. $ and leg two of the right triangle is increasing at the rate of $ \ 7 \ in/sec. $ At what rate is the triangle's

$ \ \ \ \ $ a.) hypotenuse changing

$ \ \ \ \ $ b.) perimeter changing

$ \ \ \ \ $ c.) area changing

when leg one is $ \ 8 \ in.$ and leg two is $ \ 6 \ in.$ ?Click HERE to see a detailed solution to problem 3.

- PROBLEM 4 : The radius of a circular oil slick on the surface of a pond is increasing at the rate of $ \ 10 \ meters/min. $ At what rate is the circle's

$ \ \ \ \ $ a.) circumference changing

$ \ \ \ \ $ b.) area changing

when the radius of the oil slick is $ \ 20 \ m.$ ?Click HERE to see a detailed solution to problem 4.

- PROBLEM 5 : A big block of ice is in the shape of a perfect cube. As it melts, each edge of the cube is decreasing at the rate of $ \ 2 \ cm/min. $ At what rate is the ice cube's

$ \ \ \ \ $ a.) surface area changing

$ \ \ \ \ $ b.) volume changing

when the edge of the ice cube is $ \ 80 \ cm.$ ?Click HERE to see a detailed solution to problem 5.

- PROBLEM 6 : A ladder 13 feet long is leaning against a high wall. If the base of the ladder is pushed toward the wall at the rate of $ \ 2 \ ft/sec. $, at what rate is the top of the ladder moving up the wall when the base of the ladder is

$ \ \ \ \ $ a.) 5 feet

$ \ \ \ \ $ b.) 1 foot

from the wall ?Click HERE to see a detailed solution to problem 6.

- PROBLEM 7 : The radius of a large sphere is increasing at the rate of $ \ 3 \ ft/hr. $ At what rate is the sphere's

$ \ \ \ \ $ a.) surface area changing

$ \ \ \ \ $ b.) volume changing

when the radius of the sphere is $ \ 10 \ ft.$ ?Click HERE to see a detailed solution to problem 7.

- PROBLEM 8 : Consider a closed right circular cylinder of base radius $ r \ cm.$ and height $h \ cm.$ If the radius of the cylinder is increasing at the rate of $ \ 5 \ cm/hr. $ and the height of the cylinder is decreasing at the rate of $ \ 4 \ cm/hr. $ , at what rate is the sphere's

$ \ \ \ \ $ a.) surface area changing

$ \ \ \ \ $ b.) volume changing

when the radius of the cylinder is $ \ 20 \ cm.$ and the height of the cylinder is $ 12 \ cm. $ ?Click HERE to see a detailed solution to problem 8.

- PROBLEM 9 : Consider the given isosceles triangle of base 10 inches and side lengths $x$ inches. If $x$ is increasing at the rate of $ \ 4 \ in/min. $, at what rate is the triangle's

$ \ \ \ \ $ a.) perimeter changing

$ \ \ \ \ $ b.) height changing

$ \ \ \ \ $ c.) area changing

when $ x=13 \ in. $ ?Click HERE to see a detailed solution to problem 9.

- PROBLEM 10 : Consider the given closed rectangular box with dimensions $x$ feet by $y$ feet by $z$ feet. Assume that $x$ is increasing at the rate of $ \ 4 \ ft/hr. $, $y$ is decreasing at the rate of $ \ 6 \ ft/hr. $, and $z$ is increasing at the rate of $ \ 3 \ ft/hr. $ At what rate is the box's

$ \ \ \ \ $ a.) surface area changing

$ \ \ \ \ $ b.) volume changing

$ \ \ \ \ $ c.) main diagonal changing

when $ x=5 \ ft. $, $ y=4 \ ft. $, and $ z=6 \ ft. $ ?Click HERE to see a detailed solution to problem 10.

- PROBLEM 11 : The volume of a large spherical balloon is increasing at the rate of $ \ 64 \pi \ meters^3/hr. \approx 201.06 \ meters^3/hr. $ At what rate is the balloon's surface area changing when the radius of the balloon is $ \ 2 \ m. $ ?
Click HERE to see a detailed solution to problem 11.

- PROBLEM 12 : The surface area of a cube is increasing at the rate of $ \ 600 \ in^2/hr.$ At what rate is the cube's volume changing when the edge of the cube is $ \ 10 \ in. $ ?
Click HERE to see a detailed solution to problem 12.

- PROBLEM 13 : Consider the given right triangle with legs of length $x \ cm.$ and $y \ cm.$ and angle $ \theta $ radians.
If $x$ is decreasing at the rate of $ \ 3 \ cm/min. $ and $y$ is increasing at the rate of $ \ 4 \ cm/min. $, at what rate is angle $\theta$ changing when $ x=5 \ cm. $ and $ y=2 \ cm. $ ?

Click HERE to see a detailed solution to problem 13.

- PROBLEM 14 : You are sitting $x \ ft.$ from a wall and watching a movie screen which is 10 feet high and is 6 feet above the floor. Your viewing angle is $ \theta $ radians. (See the side view diagram below.)

a.) Write your viewing angle $ \theta$ as a function of $x$.

b.) If x is increasing at the rate of $ \ 10 \ ft/min.$, at what rate is $ \theta $ changing when

$ \ \ \ \ \ \ i.) \ x=8 \ ft. $ ?

$ \ \ \ \ \ \ i.) \ x=20 \ ft. $ ?Click HERE to see a detailed solution to problem 14.

- PROBLEM 15 : Car B starts 30 miles directly east of car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. At what rate is the distance between the cars changing after

$ \ \ \ \ \ \ $ a.) $ t = 1/5 \ hr. $ ?

$ \ \ \ \ \ \ $ b.) $ t = 1/3 \ hr. $ ?

Click HERE to see a detailed solution to problem 15.

- PROBLEM 16 : An open right circular conical tank (vertex down) has height 10 meters and base radius 8 meters. Water begins flowing into the tank at the rate of $ \ \pi \ meters^3/min. $ At what rate is the depth $h$ of the water in the tank changing when

$ \ \ \ \ \ \ $ a.) $ h = 1 \ m. $ ?

$ \ \ \ \ \ \ $ b.) $ h = 9 \ m. $ ?Click HERE to see a detailed solution to problem 16.

- PROBLEM 17 : An open hemispherical tank has radius 13 feet. Oil begins flowing into the tank in such a way that the depth $h$ of the oil in the tank changes at the rate of $ 3 \ ft/hr. $ At what rate is the top circular surface area of the oil changing
when the depth of oil is

$ \ \ \ \ \ \ $ a.) $ h = 1 \ ft. $ ?

$ \ \ \ \ \ \ $ b.) $ h = 8 \ ft. $ ?Click HERE to see a detailed solution to problem 17.

- PROBLEM 18 : Car 1 starts on the graph of $ y=e^x $ at the point $ (0,1) $, and car 2 starts on the graph of $ y=3x-2 $ at the point $ (0,-2) $ and distance is measured in miles. If both cars start moving to the right at the same time in such a way that $ \displaystyle{ dx \over dt } = 1 \ mile/min. $, at what rate is the distance between the cars changing when

$ \ \ \ \ \ \ $ a.) $ t = 1 \ min. $ ?

$ \ \ \ \ \ \ $ b.) $ t = 3 \ min. $ ?Click HERE to see a detailed solution to problem 18.

- PROBLEM 19 : You are standing 12 feet from the base of a 200-ft. cliff. As a boulder rolls off the cliff, you begin running away at $ 10 \ ft/sec. $ At what rate is the distance between you and the boulder changing after

$ \ \ \ \ \ \ $ a.) $ t = 1 \ sec. $ ?

$ \ \ \ \ \ \ $ b.) $ t = 3 \ sec. $ ?Click HERE to see a detailed solution to problem 19.

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

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Duane Kouba ...
October 24, 2019