Question

Suppose you have 100 ft of string to rope off a rectangular section for a bake sale at a school fair. The function A = -x^2 + 50x gives the area of the section in square feet, where x is the width in feet. What width gives you the maximum area you can rope off. What is the maximum area? What is the range of the function?

Answer 1

I have no clue but welcome to open study

Answer 2

Have you considered finding the vertex of your parabola?
The vertex is the max of your parabola since your parabola is open down and we know this because the number in front of x^2 is negative.

Answer 3

No - how would I find the vertex?

Answer 4

You can complete the square.
Or there might be a formula your teacher shared with you. I can derive it for you but it would be nice if you learned to also complete the square...
one sec... while I type

Answer 5

err.. .I was trying to find in earlier post so I wouldn't have to type it all again...
\[y=ax^2+bx+c \\ y-c=ax^2+bx \\ \frac{y}{a}-\frac{c}{a}=x^2+\frac{b}{a}x \\ \frac{y}{a}-\frac{c}{a}+(\frac{b}{2a})^2=x^2+\frac{b}{a}x+(\frac{b}{2a})^2 \\ \frac{y}{a}-\frac{c}{a}+(\frac{b}{2a})^2=(x+\frac{b}{2a})^2 \\ \frac{y}{a}-\frac{c}{a}+\frac{b^2}{4a^2}=(x+\frac{b}{2a})^2 \\ y-c+\frac{b^2}{4a}=a(x+\frac{b}{2a})^2 \\ y=a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}\]

Answer 6

the vertex happens at x=-b/(2a)

Answer 7

Thank you very much! I got 25ft., 625 ft^2, 0 < A < or equal to 625

Answer 8

right the max area is given by the width 25ft
and the max area is given by 625ft^2

Answer 9

I would include 0 in your range because area can be zero if there is nothing there you know and just a line of 100ft of string

Answer 10

good job