Can you please help me with these maths question

1. Dad plans to surround a rectangular garden making use of the existing fence as one of the sides. Suppose 100m of fencing are available and let x be the width of the garden.

a) find the largest garden that dad can surround

Can you please help . Thank you

did you mean by largest area of the garden?
if so, then let's start from the given information..

Total perimeter = 100m
width = x,
length
= (100-2x)/2 (100 minus two width, then divide 2 to get 1 side of length)
= 50 - x

Total Area of the garden
= x (50 - x)
= 50x - x^2

let A = Area,
when A is maximum or mininum, dA/dx = 0
dA/dx = 50 - 2x
50 - 2x = 0
50 = 2x
x = 25
therefore, when x = 25, area is maximum...
(Check if x = 25 is really maximum by doing second derivative)
d^2y / dx^2 = -2 < 0 (when 2nd order derivative is negative, then it is at maximum point)

substitute x = 25, back to the original function.
A = 50x - x^2
A = 50(25) - (25^2)
A = 625 m^2

therefore, maximum area = 625m^2

hope this helps...

It depends on the length of the existing fence but since its a rectangle and the side opposite the existing fence has to be the same. So subtract the existing fence from 100 and halve the remaining value, and there you have the dimensions of the garden. If x is existing fence, (100-x) divided by 2= the other side. Hope it helps