A rectangular window is surmounted by a semicircular window that is stained. Thestained glass lets throughwindow is 24 feet. What dimensions of the window will maximize the amount of light thatwill come through the window?as much of the light as the clear glass. The perimeter of theUse the following steps. They can be used to complete many applied optimization prob-lems.Understand the problem! What information is given? What information is asked for?Can you tell that the given information does somehow determine the asked-for infor-mation? Very often this step will requirediagram.or at least be aided bya picture or|Start to set up a pure math problem. In this step, you set up a pure mathematicalcontext that models the problem. Name and define all of the variables you'll be using.Very often this step will requirethat captures the mathematical content of the picture you have already drawn; somevariables can be defined by simply labeling your diagram.or at least be aided byan abstract diagramContinue to formulate the pure math question. In this step, determine what youare trying to maximize or minimize, and write it as an expression of the variablesyou have named. (If you can't do this, consider if you need to have more variables.)Very of en the auestion will be something like "For what value of y and v is A Continue to formulate the pure math question. In this step, determine what youare trying to maximize or minimize, and write it as an expression of the variablesyou have named. (If you can't do this, consider if you need to have more variables.)Very often the question will be something like "For what value of x and y is A=xylargest?".1Use given information to refine your question. If you have multiple variables in step3, this amounts to finding an equation (or equations) relating your variables ("con-straints"). Often this equation is the result of geometrical relationships in your di-agram. Use your constraint (s) to find a function of one variable to optimize and adomain for that function. Your refined question will be something like "For whatvalues of x in [0, 10] is A(x) = (10 – x) largest?"Solve the pure math problem. Generally, you differentiate to find critical numbers. Thenyou analyze your function on its domain to find the location of its maximum or min-imum value. Remember that we have made tables of function values (for continuousfunctions on closed intervals) or graphs to help determine maximum and minimumvalues. For functions that are not continuous or intervals that are not closed, we mayneed to use information about the shape of the function from the first and secondderivative to determine if we are guaranteed a maximum or minimum value.Answer the question(s) that were asked. You might have discovered that the maxi-mum is 25 and occurs when x =5, for example. You need to reformulate the answerin the context of the problem: for example, “the room has maximum area when it is a5 feet by 5 feet square, giving an area of 25 ft²."Check that your answer (s) make sense. For example, "the room has maximum areawhen it is 0 feet by 0 feet square" does not make sense if it is possible to have a roomwith a positive area.

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Description: A rectangular window is surmounted by a semicircular window that is stained. Thestained glass lets throughwindow is 24 feet. What dimensions of the window will maximize the amount of light thatwill come through the window?as much of the light as the

The perimeter of the window is 24. The perimeter of rectangle region + perimeter of semicircle…

Answered: A rectangular window is surmounted by a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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A rectangular window is surmounted by a semicircular window that is stained. The
stained glass lets through
window is 24 feet. What dimensions of the window will maximize the amount of light that
will come through the window?
as much of the light as the clear glass. The perimeter of the
Use the following steps. They can be used to complete many applied optimization prob-
lems.
Understand the problem! What information is given? What information is asked for?
Can you tell that the given information does somehow determine the asked-for infor-
mation? Very often this step will require
diagram.
or at least be aided by
a picture or
|
Start to set up a pure math problem. In this step, you set up a pure mathematical
context that models the problem. Name and define all of the variables you'll be using.
Very often this step will require
that captures the mathematical content of the picture you have already drawn; some
variables can be defined by simply labeling your diagram.
or at least be aided by
an abstract diagram
Continue to formulate the pure math question. In this step, determine what you
are trying to maximize or minimize, and write it as an expression of the variables
you have named. (If you can't do this, consider if you need to have more variables.)
Very of en the auestion will be something like "For what value of y and v is A

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Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated