Question 616703: Question: Determine the dimension of a rectangle with a perimeter of 40cm and the greatest possible area.
Book's answer is: 10cm by 10cm
I do not get how can you get 10cm by 10cm, if it is a rectangle.
Please help me! Thank you.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
A square is just a special case of a rectangle. There is nothing wrong with calling a square a rectangle any more than it is wrong to call a square a rhombus, parallelogram, or just a plain old quadrilateral.
Having said that, given a rectangle with length  , width  , perimeter  , and area  , we can first describe the length in terms of the width and the perimeter:
Substituting this into the area formula, we create a function for area in terms of width:
\ =\ \frac{Pw\ -\ 2w^2}{2})
Rearranging in to standard quadratic form:
\ =\ -w^2\ +\ \frac{P}{2}w)
Recognizing that this second degree polynomial function in  graphs to a parabola, opening downward because of the negative lead coefficient, that must have a vertex as a maximum function value.
Using the fact that \ =\ ax^2\ +\ bx\ +\ c) has a vertex at  , we can find the vertex of our function:
}\ =\ \frac{P}{4})
This means that the maximum of the function, that is to say the maximum area rectangle for any given perimeter is a rectangle with a width one-fourth of the perimeter. If the width is one-fourth, two times the width has to be one-half of the perimeter, leaving one-half of the perimeter to account for two times the length, hence the length must also be one-fourth of the perimeter.
John
My calculator said it, I believe it, that settles it
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