Calculus 1-Related Rate Problems, guaranteed 100% Pass

1


Related Rate Problems



Related rate problems are problems where you are told about the rate at which
some quantity (or quantities) is (are) changing (i.e., you are given the value of a
derivative of a quantity with respect to time) and are asked how the rate (i.e., a
different derivative with respect to time) of some other quantity is changing.



Steps to Solve a Related Rate Problem:

1. Draw a picture if possible


2. Assign letters to quantities that are changing (these are functions of time)


3. Write down the rates that are given in the problem (these are derivatives with
respect to time) and the rate you want to find.


4. Find an equation relating the quantities whose rates you know with the
quantities whose rate you want to know (often this equation comes from the
pythagorean theorem, similar triangles, or volume/area formulas).


5. Differentiate the equation you just wrote with respect to time (𝑡).


6. Substitute values for the known rates or values of the quantities themselves
into the differentiated equation.


7. Solve for the rate (a derivative with respect to time) you want to know.

, 2


Ex. Air is being pumped into a spherical balloon at a rate of 20 cu.in./min. What
is the rate of change of the radius at the moment the diameter is 10 in.?




𝑉(𝑡) = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑟(𝑡) = 𝑟𝑎𝑑𝑖𝑢𝑠
𝑑𝑉
= 20 𝑐𝑢. 𝑖𝑛./𝑚𝑖𝑛.
𝑑𝑡
𝑑𝑟
=? at diameter= 10 𝑖𝑛. ⟹ 𝑟 = 5𝑖𝑛.
𝑑𝑡



4 3
𝑉 (𝑡) = 𝜋(𝑟(𝑡)) ; Equation relates 𝑉(𝑡) to 𝑟(𝑡). Differentiate:
3

𝑑𝑉 2 𝑑𝑟 𝑑𝑉
= 4𝜋(𝑟(𝑡)) ; Now plug in known values: = 20; 𝑟 = 5.
𝑑𝑡 𝑑𝑡 𝑑𝑡
𝑑𝑟 𝑑𝑟
20 = 4𝜋(5)2 ( ); Now solve for .
𝑑𝑡 𝑑𝑡
𝑑𝑟
20 = 100𝜋( )
𝑑𝑡
1 𝑑𝑟
5𝜋
𝑖𝑛./𝑚𝑖𝑛 =
𝑑𝑡
.