SOLUTION: A rectangle has one vertex on the line y = 8 – x (x > 0), another at the origin, one on the positive x-axis, and one on the positive y-axis. Express the area A of the rectangle as

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: A rectangle has one vertex on the line y = 8 – x (x > 0), another at the origin, one on the positive x-axis, and one on the positive y-axis. Express the area A of the rectangle as       Log On


   

Question 1055220: A rectangle has one vertex on the line y = 8 – x (x > 0), another at the origin, one on the positive x-axis, and one on the positive y-axis. Express the area A of the rectangle as a function of x. Find the largest area A that can be enclosed by the rectangle.
Answer by ikleyn(52781) About Me  (Show Source):
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A rectangle has one vertex on the line y = 8 – x (x > 0), another at the origin, one on the positive x-axis,
and one on the positive y-axis. Express the area A of the rectangle as a function of x.
Find the largest area A that can be enclosed by the rectangle.
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1.  The area of a rectangle is  A = x*(8-x),  or  A = -x^2 + 8x.

    Simply because one dimension is x,  while the other dimension is  y = (8-x).


2.  The maximum of the quadratic function  A = -x^2 + 8x  is at

    x = -b%2F%282a%29 = -8%2F%282%2A%28-1%29%29 = 4.

    Then  x = 4, y = 8-x = 4  and  A = 4*4 = 16 square units  is the maximal area.

See the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.


The referred lessons are the part of this online textbook under the topic "Finding minimum/maximum of quadratic functions".