Change of perimeter
Change of area
Change of perimeter
Change of area
Example 3) A right triangle has sides of 30 and 40 inches whose sides are changing. Write formulas for the area
of the triangle and the hypotenuse of the triangle and how fast the area and hypotenuse are changing
Area Change of area
Hypotenuse
Change of hypotenuse
a) the short side is increasing at 3 in./sec
b) the short side is increasing at 3 in./sec and the long
and the long side is increasing at 5 in/sec.
side is decreasing at 5 in./sec.
Change of area Change of hypotenuse
Change of area
Change of hypotenuse

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Stu Schwartz
Example 4. A right circular cylinder has a height of 10 feet and radius 8 feet whose dimensions are changing.
Write formulas for the volume and surface area of the cylinder and the rate at which they change.
Volume Change of volume
Surface area
Change of surface area
a) the radius is growing at 2 feet/min and
b) the radius is decreasing at 4 feet/min and the
the height is shrinking at 3 feet/min.
the height is increasing at 2 feet/min.
Change of volume Change of surface area
Change of volume Change of surface area
To solve related rates problems, you need a strategy that always works. Related rates problems always can be
recognized by the words “increasing, decreasing, growing, shrinking, changing.” Follow these guidelines in solving a
related rates problem.
1. Make a sketch. Label all sides in terms of variables even if you are given the actual values of the sides.
2. You will make a table of variables. The table will contain two types of variables - variables that are constants
and variables that are changing. Variables that never change go into the constant column. Variables that are
a given value only at a certain point in time go into the changing column. Rates (recognized by “increasing”,
“decreasing”, etc.) are derivatives with respect to time and can go in either column.
3. Find an equation which ties your variables together. If it an area problem, you need an area equation. If it is a
right triangle, the Pythagorean formula may work or gerenal trig formulas may apply. If it is a general triangle,
the law of cosines may work.
4. You may now plug in any variable in the constant column.
Never
plug in any variable in the changing column.
5. Differentiate your equation with respect to time. You are doing implicit differentiation with respect to
t
.
6. Plug in all variables. Hopefully, you will know all variables except one. If not, you will need an equation which
will solve for unknown variables. Many times, it is the same equation as the one you used above. Do this work
on the side as to not destroy the momentum of your work so far.
7. Label your answers in terms of the correct units (very important) and be sure you answered the question asked.

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- Fall '16
- Guzman
- Calculus, Derivative, Stu Schwartz