Question

The length plus the width of a rectangle is 10

. Let x represent the length. The equation for the area y of the rectangle is y = x(10 – x). The y-value of the vertex gives the (maximum or minimum) value of the (length or area or width)of the rectangle

Answer 1

maximum value of the area of the rectangle is 25 units^2.

Explanation:

Let x represent the length of the rectangle. Then the width must be w = 10 - x.

Thus we have an equation for the area (x(10 - x)) = y of the rectangle.

Expanding this by performing the indicated multiplication, we get the area formula y = x(10 - x), or y = -x^2 + 10x + 0.

The coefficients of the x terms of this quadratic are {-1, 10, 0}.

The vertex is at x = -b/(2a), which here is x = -10/(2*-1), or x = 5.

The y value, representing the area of the rectangle, is y = x(10 - x), or, in this particular case, y = 5(10 - 5), or y = 25. This is the maximum possible value of the area of the rectangle.