it takes 80 feet of fencing to enclose your garden. According to your calculations,you will need 350 square feet to plant everything you want. Is your garden big enough? Explain your answer.

Question

They gave a rectangle length=40-x and width= x
I asked my mom and she also got confused.So we are stuck.

Answer 1

so far, so good. The area of such a rectangle is

x(40-x) = 40x - x^2

You want

40x - x^2 >= 350
-x^2 + 40x - 350 >= 0

Now, the roots of this function are
12.9 and 27.1

Since it is a parabola opening down, y is positive for x between the roots. So, for any value of x between 12.9 and 27.1, the garden will be more than 350 ft^2.

Or, you could have examined the equation and noted that the vertex is at x=20, where there is maximum area. f(20) = 400, more than big enough.

Just for future reference, the maximum area possible for a rectangle of given perimeter is when the rectangle is a square. So, with 80 feet of fencing, the maximum area is 400 ft^2, in a 20x20 square. So, it should have been immediately clear that there was enough fencing to enclose 350 ft^2.