Question

Bob has 50 feet of fencing to enclose a rectangular garden. If one side of the garden is x feet long, he wants the other side to be (25 – x) feet wide. What value of x will give the largest area, in square feet, for the garden?

Answer 1

this is rather complicated x=(50)^0.5=12.5
let a be area,
a=(25-x) (x)
da/dx=25-2x
dadx=0
25-2x=0
x=12.5

Answer 2

Answer:

for $x= 12.5 \ feet$ the area will be largest

Step-by-step explanation:

It is given that one side of rectangular garden is x feet

and other side is 25-x feet

Now the area of the rectangle garden is given by

$A=(25-x)x$

$A=25x-x^2$ ( we distribute x)

$A= -x^2 +25x$ ( writing quadratic equation in standard form)

A quadratic function $y=ax^2+bx+c$ with negative value of a , is a parabola with maximum value at vertex.

the x coordinate of vertex is given by

$x=-\frac{b}{2a}$

We compare $-x^2 +25x$ with  $ax^2+bx+c$

so we have $a=-1\ b=25\ c=0$

The x coordinate of vertex is given by

$x=-\frac{25}{2(-1)}$

$x=12.5$

hence for $x= 12.5 \ feet$ the area will be largest