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MORE NOTES
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Constructing a closed box. A closed box with a square base is required to have a volume of 10 cubic feet.
b) Express the amount of material (M) to make the
box as a function of the length of the square base (x).
x 1 base, a 2 x 2 base, and a 5 x 5 base.
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c) G ap y = M(x). Whoa ... hold up, A-~ed ~aphmg calc for th1s!
Show your window and make a general sketch below.
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d) For ~at va ues ofx is the surface area
What is the surface area?
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=(x,y) be a point on the graph off ( x).
a) Express the distance (d) from P to the point (I, 0) as a function ofx.
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b) Sketch on the diagram where you think the distance will be the smallest.
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c) Why would it be silly for me to ask where dis the largest?
d) Graph d(x) on your calc. Sketch below.
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e) For what value(s) ofx is d smallest?
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I 0. Graph
f ( x) = x2 -
1. Let P = (x,y) be a point on the graph of
f ( x).
a) Express the distance (d) from P to the origin as a function ofx.
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b) What is the distance if x
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c)Sketch on the diagram where you think the distance will be the smallest.
e) For what value(s) ofx is d smallest?
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II. A rectangle is inscribed in a semicircle of-radius 2. Let P
the circle. See the figure.
a) Express the area A of the rectangle
as a function of x.
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(x,y) be the point in quadrant I that is the vertex of the
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b) Express the perimeter p of the rectangle
as a function of x.
A -= bh
A(:~-)= ;)x~l{-~
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c) Graph A(x) . For what value ofx is A largest?
rectaiJ~le an ~n
d) Graph p(x). For what value ofx is p largest?
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MORE Word Problems
12.
Gett ing from an Island to To wn An island is 2 miles from tl1e nearest point P on a straight shore line. A town is 12 miles
down the shore from P.
a)
If a person can ro w a boat at an average speed of 3 miles per hour and the same person can wa lk S miles per hour,
express the time T that is takes to go from the island to town as a function of the distance x from P to where the
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b)
What is the domain ofT?
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Lowest x (for the problem)
Highest x (in theory)
Lowest x (in theory)
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Domain for the function
Domain for
How long wi~take to travel from
the island to town if the person goes directly?
Sketch this path.
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d) How long will it take to travel from the island
to the town if you go straight to the shore and
then to the town? Sketch this path.
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Graph the funct ion T=T(x) on your calculator.
Sketch below. Show the window.
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Why are the domains different?
c)
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Highest reasonable x for the problem
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f) Use the CALC menu to find the value ofx that results
in the least time. Sketch this path .
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13. Susie has moved into a shan ty in the woods with no plumbing. The local contractor inform s her that it wi ll cost $3.5/ft to lay piping
along the road and $50/ ft to lay piping across 1~. He1· house is 100ft from the closest point P on a stra ight road and the
·
nearest piping is 350ft down the road from P ~"'-"'OlD~ •
a.
b.
Compute the cost if x = 30 feet.
Sketch this s ituation.
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c) G@I?..\l.£~x).
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d) What value of x results
in the least cost? What is
the least cost?
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14. A manufacturer of playpens maJe square mo el that can be opened at one comer and attached to a wall. If each side is 3 feet in
length, the open configuration doubles the available play area from 9 square feet to 18 square feet. If we place hinges at the outer
comers (like the picture) we can increase the area again.
Top views of three configurations.
a)
Build a model that expresses the area A of the hinged configuration as a function of the distance
x between the two parallel sides.
b)
Find the domain of A(x) for this problem.
d)
For what value ofx is the area the largest? What is the maximum area?
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c) Find A ifx
=
5.
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