Download Worksheet - UCSB Math

Student Name(s):
Optimization
Computer Lab
In this computer lab, you will do some optimization problems having to do with the
volume of a cone. Start by clicking Evaluation->Evaluate Notebook. Then read the
instructions below, and start to answer the questions.
Instructions: For each problem, you are to find the maximum volume of a cone given
different constraints. You should give a decimal approximation to the answers based on what
you observe in the modules (you do not need to solve the problems using calculus, except in
the first problem).
To make use of the modules, you will need to write down the constraint equation, and
then solve for either r or h in terms of the other variable. Then enter this expression into
the appropriate module, and use the slider to observe what the maximum volume is.
1. In this question we will find the maximum volume of a cone whose height and radius
add to 10.
(a) Write down the constraint as an equation relating r and h.
(b) Solve for either r or h. That is, either write h as a function of r or vice versa.
(c) If you solved for h, then use the module “Cone: control r”, and enter the function of
r in the input field. If you solved for r, use the “Volume of a Cone: r as a function
of h” module, and enter the function of h in the input field. Sketch the graph that
you obtain, and explain in one sentence why the h-intercepts or r-intercepts are
what they are.
6y
x
?
-
(d) Use the graph and slider to estimate the maximum volume, and say what height
and radius yield this volume.
V =
h=
r=
(e) Redo part (d) using calculus to find the exact values (not decimal approximations)
of r and h which maximize the volume given the constraint. Remember, the volume
1
of a cone is V = πr2 h.
3
2. The slant height L of a cone is the distance from a point on the edge of the base to
the tip. A visualization is included at the bottom of the Mathematica notebook. Write
down an equation relating the slant height L, the height h, and the radius r of a cone.
3. In this question, we will find the maximum volume of a cone whose slant height is 10.
(a) Sketch a cone with slant height 10, radius r and height h. Then write down the
constraint in terms of r and h only.
(b) Solve for either r or h in the constraint equation.
(c) As before, use one of the two modules to enter h as a function of r or vice versa.
Then sketch the graph you obtain.
6y
x
?
-
(d) Use the graph and slider to estimate the maximum volume, and say what height
and radius yield this volume.
V =
h=
r=
(e) Explain in one sentence why you think the maximum volume occurs when r is
greater than h.
4. The surface area of the curved face of a cone is πrL, where r is the radius and L is
the slant height. We will find the maximum volume of a cone whose curved face has
surface area 200.
(a) Write down the constraint in terms of r and h. (You will have to substitute for L)
(b) Use algebra to solve for either r or h (it is much easier to solve for h as a function
of r in this example).
(c) Sketch the graph that you obtain. It should appear that one of the h-intercepts is
8. Is it exactly 8, or is it something else?
6y
x
?
-
(d) Compare the cones when r = 1 to when r = 7. Explain why the curved surfaces of
these two very differently shaped cones have the same surface area.
(e) Use the graph and slider to estimate the maximum volume, and say what height
and radius yield this volume. Also, say what the slant height L is for this cone.
V =
h=
r=
L=
5. The total surface area of a cone is πrL + πr2 (the curved face plus the base). In this
question, we find the maximum volume of a cone whose total surface area is 200.
(a) Write down the constraint in terms of r and h. (You will have to substitue for L).
(b) Use algebra to solve for either r or h (it is much easier to solve for h as a function
of r in this example).
(c) Use the graph and slider to estimate the maximum volume, and say what height
and radius yield this volume. Also, say what the slant height L is for this cone.
V =
h=
r=
L=