Section 5.4: Optimization
Example: A rectangular area is to be enclosed by a fence and divided into two equal parts,
with the fence running along the division parallel to one of the sides. If the total area is 384
square feet, find the dimensions that will minimize the total length of the fence. How much
fencing is required?
Example: Find the largest possible area of a right triangle whose hypotenuse is 4 cm long.
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Example: Find the dimensions of the rectangle of largest area that has its base on the x-axis
and its other two vertices above the x-axis and lying on the parabola y = 8 − x2 .
Example: Find the point on the line y = 2x + 3 that is closest to the origin.
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Example: If 1200 cm2 of material is available to make a box with a square base and open
top, find the largest possible volume of the box.
Example: A can is to be made to hold 1 L (1000 cm3 ) of oil. Find the dimensions that will
minimize the cost of the metal to manufacture the can.
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Example: (Epidemiology of Measles)
To model the development of measles, we use the measles pathogenesis function
f (t) = −t(t − 21)(t + 1),
where t is measured in days and f (t) represents the number of infected cells per milliliter of
plasma. What is the peak infection time for the measles virus?
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