Math 151: Precalculus Optimization
To determine the greatest or least, the biggest or smallest, the oldest or youngest, the
best or worst, the heaviest or lightest, the cheapest or most expensive ... these are
problems of finding the optimum (maximum or minimum) quantity. Such word problems
are called “Optimization Word Problems”.
Optimization (Applied Max/Min Word Problem) Steps:
Define non - x and y variables for both independent and dependent quantities.
Sketch a labeled diagram, if possible.
Write all equations for the quantity to be maximized or minimized (optimized)
Eliminate variables.
Rewrite the optimizing function in terms of x and y.
Determine the domain of the function.
Identify the desired maximum or minimum function value by graphing the function on
a CAS and use the minimum or maximum command.
8. Answer the question completely.
1.
2.
3.
4.
5.
6.
7.
Example - The United States Parcel Service has contracted you to design a cylindrical
container that has a volume of 100 cubic inches and a minimum surface area. What are
the dimensions of such a cylindrical container?
1. Define variables. Identify all dependent quantities and all independent quantities,
and define them using NON x and y variables. We will use x and y eventually, but
NOT YET. This is important unless you like getting confused and getting things
wrong.
Independent Quantities
Let r be the radius of the cylinder.
Let h be the height of the cylinder.
2. Sketch and label a diagram, if possible.
Dependent Quantities
Let V be the volume of the cylinder.
Let SA be the surface area of the
cylinder.
r
h
3. Write all equations. Write an equation for each dependent variable in terms of all of
the independent variables. Do NOT substitute any numbers or other relationships in
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Math 151: Precalculus Optimization
for variables at this stage yet. Later on, when you are more comfortable, you can
combine steps, but for now, stick with a guaranteed, “one-size-fits-all” approach:
( )
Volume: V r, h = πr 2h , where r is the radius and h is the height of the cylinder.
( )
Surface Area: SA r, h = 2πr 2 + 2πrh (this includes the top and bottom of the
cylinder).
Note that the dependent quantities of V and SA are written using function notation
with TWO input variables. This is important, because it reminds us of what the
independent quantities are that will be involved in calculating the volume and surface
area.
4. Eliminate variables. This next step is where we reduce the number of variables to
ONE dependent and ONE independent. Look for information in the word problem
that specifies the value of ONE of the dependent quantities; this is the dependent
quantity that we will eliminate. Also, confirm that the other dependent variable is
the quantity to be optimized (made biggest or smallest); this is the dependent
quantity that we will keep. Decide from the word problem which independent quantity
to keep based on what you are asked to solve for. If both independent variables are
required, choose the independent variable that makes the algebra simplest.
We are given that the volume must be 100 in3 : V = 100 . Furthermore, we want to
minimize the surface area.
So, V is the dependent variable to eliminate and SA is the one to keep.
This specific given information allows us to relate radius and height so that we can
deal with only one independent variable, either radius or height.
( )
Therefore:
V r, h = πr 2h
Since we need to solve for both radius and
100 = πr 2h
height, and the algebra is simpler if we
100
100
keep r, let’s eliminate h.
πr 2
πh
The optimizing equation for one dependent variable, surface area, in terms of one
independent variable, radius, is:
h=
( )
SA (r ) = 2πr
SA (r ) = 2πr
or r =
SA r, h = 2πr 2 + 2πrh
2
2
+ 2πr ⋅
+
100
πr 2
by substituting h =
100
πr 2
200
r
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Math 151: Precalculus Optimization
5. Rewrite the optimizing equation in terms of x and y.
200
Let y be the surface area and x be the radius: y = 2π x 2 +
x
6. Determine all domain restrictions. If the domain is a closed interval, we will need to
check the function behaviour at the endpoints. Some people completely ignore this
step. Don’t!  Always determine the domain, even if it is all real numbers.
Consider all independent variables. Use common sense and all information from the
word problem.
Dimensions must be positive: r > 0 and h > 0
∴ r > 0 and
100
πr 2
>0
∴ r > 0 and r 2 > 0
Therefore the domain is r > 0 .
Note that there are no endpoints to consider for candidates for the maximum or
minimum Surface Area.
7. Identify the desired maximum or minimum function (y) value.
a) Precalculus method: Sketch the graph of the function on the domain and identify
the desired maximum or minimum point as
dictated by the y - coordinate (NOT the x –
coordinate).
y = 2π x 2 +
200
x
,x > 0
Minimum point at
(x, y ) = (r, SA) = (2.515,119.265) .
8. Answer the question. Interpret the maximum or minimum ordered pair values
correctly.
The cylinder whose volume is 100 in3 has a minimum surface area of 119.265 in2 when
the radius is 2.515 in and the height is h =
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100
πr 2
=
(
100
π 2.515
)
2
= 5.032 in .
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Math 151: Precalculus Optimization
Answer the following on looseleaf.
Exercise 1: The sum of two nonnegative numbers is 20. Determine the numbers if the
sum of their squares is
a) as small as possible.
b) as large as possible.
Exercise 2: An open-top box is to be made by cutting congruent squares from the
corner of a 20-inch by 25-inch sheet of tin and bending up the sides. How large should
the corners be to make the box hold as much as possible? What volume results?
⎛3 ⎞
Exercise 3: How close does the curve y = x come to the point ⎜ , 0⎟ ?
⎝2 ⎠
Exercise 4: Bianca is in a boat on the water 2 miles perpendicular from lifeguard station
A that is on the shore. She wishes to reach a second lifeguard station B on the shore
that is 6 miles down the beach from lifeguard station A as quickly as possible. Bianca’s
rowing speed is 2 miles per hour and her walking speed along the beach is 5 miles per
hour.
a) Sketch diagrams for each of the 3 different scenarios as to how Bianca can propel
herself to lifeguard station B from her boat in the water, two of which are limiting
cases for the third case.
b) Locate the point along the beach to which Bianca should row to get from her point in
the water to lifeguard station B as quickly as possible.
Exercise 5: A 20-foot long trough has trapezoidal ends with dimensions shown in the
diagram. Determine the optimal value of the angle θ such that this trough will hold the
most amount of water.
1 ft
θ
θ
1 ft
1 ft
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20 feet
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