Question

Beth wants to plant a garden at the back of her house. She has 32m of fencing. The area that can be enclosed is modelled by the function A(x) = -2x2 + 32x, where x is the width of the garden in metres and A(x) is the area in square metres. What is the maximum area that can be enclosed?
Please help :(

Answer 1

Answer:

The maximum area that can be obtained by the garden is 128 meters squared.

Step-by-step explanation:

A represents area and we want to know the maximum.

$A(x)=-2x^2+32x$ is a parabola.  To find the maximum of a parabola, you need to find it's vertex.  The y-coordinate of the vertex will give us the maximum area.

To do this we will need to first find the x-coordinate of our vertex.

$x=\frac{-b}{2a}{/tex] will give us the x-coordinate of the vertex.Compare [tex]-2x^2+32x$ to $ax^2+bx+c$ then $a=-2,b=32,c=0[tex].So the x-coordinate is [tex]\frac{-(32)}{2(-2)}=\frac{-32}{-4}=8$.

To find the y that corresponds use the equation that relates y and x.

$y=-2x^2+32x$

$y=-2(8)^2+32(8)$

$y=-2(64)+32(8)$

$y=-128+256$

$y=128$

The maximum area that can be obtained by the garden is 128 meters squared.