Which of these numbers are rational? (Choose all that apply)
1 answer:
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Answer:
(a) x<8
(b)
(c)
(d)
Step-by-step explanation:
<u>Optimization </u>
It is the procedure to find the set of values for the variables of a function such that it reaches a maximum or a minimum value. If equalities are given as relationships between the variables, then the derivative is a suitable method to find the critical points or candidates for extrema values.
The problem at hand is about a geometric maximization, given some dimensional conditions. First, we have a rectangular piece of cardboard measuring 24 x 16 inches. A box is to be made out of that cardboard by cutting equal size squares from each corner and folding up the sides of length x.
(a)
When we do so, the base of the box will have dimensions
Since the new width of the base must be positive, then
which poses the restriction
The same situation happens with the length
x<8
Since this last condition is more restrictive than the first, we state that x must be less than 8
(b) The volume of the box is the product of the area of the base by the height x
Operating
(c)
To find the maximum volume, we take the first derivative of V:
Equating to zero to find the critical points
Dividing by 4:
The roots of the equation are:
Since x=10.19 is out of the restrictions found in part a, the only valid solution is
We must test if the critical point is a maximum or a minimum, by computing the second derivative
Since the second derivative is negative, the value is a maximum
(d) We must find all the values of x such as v>288:
Rearranging
Simplifying by 4
Factoring
We found three real and positive roots for the third-degree polynomial
The function is positive when
or
The only interval lying into the valid values of x is