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Related Rates Activity: Filming Rocket
A television camera is positioned 4000 ft from the base of rocket launching pad. The angle
of elevation of the camera has to change at the correct rate in order tokeep the rocket in sight.
Suppose that the rocket's speed is 600 ft/sec at the instant that is has risen 3000 ft. How
fast is the camera's angle of elevation changing at that same moment?
Watch the accompanying video and then answer the following questions.
1. Write down specifically what the variables y, s. and 0 each represent. Make sure to
include units.
2. Rewrite the equation relating the variables as
o(t) arctan
do
Differentiate this equation implicitly. Do you obtain the same value at the "special
dt
time" t*? (Should you obtain the same value?)
3. Write an equation relating the variables using sine instead of tangent. and differentiate
d0
this equation instead. Do you get the same answer for at t*? (Should you
dt
obtain the same value?)
4. How do your answer and your work change if the rocket is instead 2250 ft above the
launchpad at £*?
5. Suppose the rocket were instead descending slowly at rate of 10 ft/sec at t = E*
What is the value of at t= t*? What is the rate of change of the angle 0(t) at
dy
dt
6. How fast is the distance from the television camera to the rocket changing at = {*?
Identify the relevant variables and form an equation relating them to answer this
question. (There are multiple possible equations!)
7. Suppose that from an elevation of 3000 ft onward, the rocket continues to rise at 600
ds
ft/sec. Will be greater at the instant when the elevation is 4000 ft than it was at
dt
3000 or less? Justify.
Optimization, II
In this set of examples and activities, we continue our discussion of optimization prob-
lems. The corresponding material in our text is Section 4.7 (Stewart).
Previously, we examined class of problems that we broadly construed as "fencing
problems" which involved (or had related interpretations of) the change in: product, like
an area. subject to constraints on an input (like perimeter). Sometimes such pr oblems are
inverted in the sense that 'perimeter may be optimized while the *area¹-like quantity
is fixed. This collection of problems to which we are referring as "fencing problems"
has multiple generalizations. One such sort is to add some other feature of length or
perimeter into the mix, such as cost per unit of length, rather than area, as the feature
to be optimized, as we will see in one of our examples. Another generalization passes
to higher dimensions, replacing 'area' by 'volume' and 'perimeter by 'surface area'
so
that, for instance, we may need to optimize surface area while volume remains fixed, or
optimize volume whilesurface area remains fixed and we'l examine sample such
problerns.
As with prior problems, keep in mind theset of "Steps for Approaching Optimization
Problems" from Section 4.7.
Example 1. An oil refinery is located on the north bank of straight river that is km
wide. A pipeline i to be constructed from the refiner to the storage tanks located on the
south bank of the river km east of the refinery. The cost of laying pipe is $400,000/km
over land to point P on the north bank and $800,000/km under the river to the tanks.
To minimize the cost of the pipeline, where should P be located?
Activity
1. Watch the video "Minimizing Cost Laying Pipe", in this module.
2. Reflect on the following questions to the video, and submit your responses result
to the dropbox, "Tuesday 3-24 Activities":
What else should we check to make this more complete solution? Are there
other sources of critical points that we should consider?
Will that change our solution? Why or why not?
Example 2. A box with square base and an open top must have volume of 32.000
cm³ Find the dimensions of the box that minimize the amount of material used.
Activity 2.
1. Watch the video. "Applied Optimization: Minimizing the Material of a Box"
2. Open the worksheet, "Spring2020CanOpti in this module. Complete the Ques-
tions 3 in the worksheet associated to this box problem.
Example 3. Suppose now that you are to design a closed-top aluminum can to be 1
liter (1000 cm³ in volume. Assume that the top, bottom, and side of the can all have
the same thickness.
Activity 3.
1. Continue with the worksheet. "Spring2020CanOpti" in this module. Complete the
Questions 1 - in the worksheet. associated to this can problem. (You will need the
Geogabra worksheet, "Closed Top Can Optimization"
Example 4. Consider the can in the previous example. Suppose now that the man-
ufacturer will take waste into account. There is no waste in cutting the aluminum for
the side of the can. but the top and bottom of radius will be cut from squares which
measure 2r units on side.
Activity 4.
1. Continue with the worksheet, "Spring2020CanOpti' in this module. Complete the
Questions 1 - in the worksheet associated to this second can problem. (You will
need the Geograbra worksheet "Closed-Top Can Optimization (with Waste)"

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