Question

Shenelle has 100100100 meters of fencing to build a rectangular garden. the garden's area (in square meters) as a function of the garden's width www (in meters) is modeled by: a(w)=-(w-25)^2+625a(w)=−(w−25) 2 +625a, left parenthesis, w, right parenthesis, equals, minus, left parenthesis, w, minus, 25, right parenthesis, start superscript, 2, end superscript, plus, 625 what is the maximum area possible?

Answer 1

Area as a function of width, w is:
a(w)=(w-25)^2+625
for maximum area, the width will be:
a'(w)=2(w-25)+0=0
solving for w we egt
2w-50=0
2w=50
w=25 m
given that the perimeter is 100, the length will be:
100=2(L+W)
solving for L we get:
L=50-W
but W=25m
hence
L=50-25=25 m
thus the maximum area will be:
A=L*W=25*25=625m^2

Answer 2

Answer:

625 sq. m.

Step-by-step explanation:

The graden's area is given as :  $a(w) = -(w-25)^2+625$

For maximization of area,we need to put first order derivative of given equation equals to 0 i.e. a'(w) should be equal to zero.  

$a'(w) = -2(w-25) = 0$

On Simplifying :

$w-25 = 0$

w = 25 meters

Since we are given that Shenelle has 100 meters of fencing to build a rectangular garden

This means perimeter is equal to 100.

Formula of perimeter of rectangle : $2(w+l) = 100$

Where l is length

w is width

Plugging w as 25

$2(25+l) = 100$

Dividing 2 in both sides

$(25+l) = 50$

$l = 25$

So, for the maximum area the length should be 25 m and width should 25 m

So, So, maximum area $=Length \times Width = 25 \times 25 = 625 m^2$

Hence the maximum possible area is 625 sq. m.