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3.4. APPLIED OPTIMIZATION
maximum? That is, finding the absolute maximum volume of a parcel is different
from finding the dimensions of the parcel that produce the maximum.
Activity 3.10.
A soup can in the shape of a right circular cylinder is to be made from two materials.
The material for the side of the can costs $0.015 per square inch and the material for
the lids costs $0.027 per square inch. Suppose that we desire to construct a can that
has a volume of 16 cubic inches. What dimensions minimize the cost of the can?
(a) Draw a picture of the can and label its dimensions with appropriate variables.
(b) Use your variables to determine expressions for the volume, surface area, and
cost of the can.
(c) Determine the total cost function as a function of a single variable. What is the
domain on which you should consider this function?
(d) Find the absolute minimum cost and the dimensions that produce this value.
⊳
Familiarity with common geometric formulas is particularly helpful in problems like
the one in Activity 3.10. Sometimes those involve perimeter, area, volume, or surface
area. At other times, the constraints of a problem introduce right triangles (where the
Pythagorean Theorem applies) or other functions whose formulas provide relationships
among variables present.
Activity 3.11.
A hiker starting at a point P on a straight road walks east towards point Q, which
is on the road and 3 kilometers from point P. Two kilometers due north of point Q
is a cabin. The hiker will walk down the road for a while, at a pace of 8 kilometers
per hour. At some point Z between P and Q, the hiker leaves the road and makes a
straight line towards the cabin through the woods, hiking at a pace of 3 kph, as pictured
in Figure 3.22. In order to minimize the time to go from P to Z to the cabin, where
should the hiker turn into the forest?
⊳
In more geometric problems, we often use curves or functions to provide natural constraints. For instance, we could investigate which isosceles triangle that circumscribes a unit
circle has the smallest area, which you can explore for yourself at http://gvsu.edu/s/9b.
Or similarly, for a region bounded by a parabola, we might seek the rectangle of largest
area that fits beneath the curve, as shown at http://gvsu.edu/s/9c. The next activity
is similar to the latter problem.
Activity 3.12.
Consider the region in the x-y plane that is bounded by the x-axis and the function
f (x) = 25 − x 2 . Construct a rectangle whose base lies on the x-axis and is centered at
3.4. APPLIED OPTIMIZATION
maximum? That is, finding the absolute maximum volume of a parcel is different
from finding the dimensions of the parcel that produce the maximum.
Activity 3.10.
A soup can in the shape of a right circular cylinder is to be made from two materials.
The material for the side of the can costs $0.015 per square inch and the material for
the lids costs $0.027 per square inch. Suppose that we desire to construct a can that
has a volume of 16 cubic inches. What dimensions minimize the cost of the can?
(a) Draw a picture of the can and label its dimensions with appropriate variables.
(b) Use your variables to determine expressions for the volume, surface area, and
cost of the can.
(c) Determine the total cost function as a function of a single variable. What is the
domain on which you should consider this function?
(d) Find the absolute minimum cost and the dimensions that produce this value.
⊳
Familiarity with common geometric formulas is particularly helpful in problems like
the one in Activity 3.10. Sometimes those involve perimeter, area, volume, or surface
area. At other times, the constraints of a problem introduce right triangles (where the
Pythagorean Theorem applies) or other functions whose formulas provide relationships
among variables present.
Activity 3.11.
A hiker starting at a point P on a straight road walks east towards point Q, which
is on the road and 3 kilometers from point P. Two kilometers due north of point Q
is a cabin. The hiker will walk down the road for a while, at a pace of 8 kilometers
per hour. At some point Z between P and Q, the hiker leaves the road and makes a
straight line towards the cabin through the woods, hiking at a pace of 3 kph, as pictured
in Figure 3.22. In order to minimize the time to go from P to Z to the cabin, where
should the hiker turn into the forest?
⊳
In more geometric problems, we often use curves or functions to provide natural constraints. For instance, we could investigate which isosceles triangle that circumscribes a unit
circle has the smallest area, which you can explore for yourself at http://gvsu.edu/s/9b.
Or similarly, for a region bounded by a parabola, we might seek the rectangle of largest
area that fits beneath the curve, as shown at http://gvsu.edu/s/9c. The next activity
is similar to the latter problem.
Activity 3.12.
Consider the region in the x-y plane that is bounded by the x-axis and the function
f (x) = 25 − x 2 . Construct a rectangle whose base lies on the x-axis and is centered at