I'm doing the rug problem that starts with rug of 9' x 8'. The weaver wants to cut of x ft from the 9' length and add it to the 8' width. I think the equation is y= (9-x)(8+x). This however creates a vertex of 72.25'. I don't see how the area can exceed 72'. I therefore expected vertex to be 72. Can't see how the weaver can create a rug of more area than he starts with.

Answer:

Step-by-step explanation:

In this problem, the original area of the rug is 9' x 8' = 72 square feet.

If the weaver cuts off x feet from the 9' length, then the new length will be (9 - x) feet.

If the weaver adds this x feet to the 8' width, then the new width will be (8 + x) feet.

So, the new area of the rug will be:

new area = new length × new width

We can represent the area of the new rug as a function of x, given by:

A(x) = (9 - x)(8 + x)

We can rewrite this equation in vertex form by completing the square. 

A(x) = -x² - x + 72

To complete the square, we need to add and subtract (1/4) to the expression - x² - x:

A(x) = -(x² + x + 1/4) + 1/4 + 72

A(x) = -(x + 1/2)² + 145/2

Now we can see that the vertex of the parabola is at the point (-1/2, 145/2). Since the coefficient of x² is negative, this is a downward-facing parabola with a maximum value at the vertex. Therefore, the maximum area of the new rug is obtained when x = -1/2, which corresponds to the center of the parabola. Plugging this value of x into the equation, we get:

A(-1/2) = -(1/2 + 1/2)²+ 145/2 = 145/2 - 1/4 = 289/2

So the maximum area of the new rug is 289/2 square feet, which is approximately 72.25 square feet.

However, note that this value is not attainable because it corresponds to cutting off -1/2 feet from one end of the original rug and sewing it onto the width of the rug, which is not physically possible. The closest attainable value of x is 0 (i.e., not cutting off any length from the original rug), which gives an area of:

A(0) = (9 - 0)(8 + 0) = 72

Therefore, the maximum area of the new rug that can be obtained by cutting off some length from one end of the original rug and sewing it onto the width of the rug is 72 square feet.