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What are the two numbers that are \(42\) apart
and whose product is minimal?
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Suppose you want to build a rectangular,
fenced-in pen along a river.
This way you only need to build three sides of the pen.
You have \(400\mathrm{ft}\) of fencing.
What’s the maximum possible area of the plot?
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What is the point on the curve \(y^2 = x\)
that is closest to the point \((0,5)?\)
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What is the minimum vertical distance
between these two parabolas?
\[
y = x^2+7x+5
\qquad\qquad
y = 2x-x^2
\]
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Using the fact that the vertex or a parabola
is the location of its global minimum/maximum
(depending on its orientation),
show that the \(x\)-coordinate of the vertex of the parabola
given by the graph of \(f(x) = ax^2+bx+c\)
is \(\frac{-b}{2a}.\)
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A straight piece of wire 40 cm long is cut into two pieces.
One piece is bent into a circle and the other is bent into a square.
How should wire be cut so that the total area of both circle and square is minimized?
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The top and bottom margins of a poster are each \(6\mathrm{cm}\),
and the side margins are each \(4\mathrm{cm}\).
If the area of the printed material on the poster
(that is, the area between the margins) is fixed at \(384\mathrm{cm}^2,\)
find the dimensions of the poster with the smallest total area.
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A water trough — like a rain gutter —
is to be made from a long strip of tin 8in wide
by taking 3in on each side of that width
and bending them up at and angle \(\theta\).
What angle \(\theta\) would maximize the cross sectional area,
and thus the amount of water, the trough holds?
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Maya is 2 km offshore in a boat
and wishes to reach a coastal village
which is 6 km down a straight shoreline
from the point on the shore nearest to the boat.
She can row at 2 km/hr and run at 5 km/hr.
Where should she land her boat to reach the village
in the least amount of time?
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A storage container is to be made in the form
of a right circular cylinder with a volume of 287ft².
Material for the top of the container costs $5 per square foot
and material for the sides and base costs $2 per square foot.
What dimensions will minimize the total cost of the container?