Differential calculus
Calculus
Level
2
A rectangular garden consists of a rectangular lawn of area 18m² surrounded by a gravel path. The path is 2m wide at the top and bottom ends of the garden and 1m wide along each side. Find the dimensions of the garden of smallest area satisfying the given specifications.
X=3 y=4
x=10 y=2
x=5 y=10
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
xy=A _ (1)
(x-2)(y-4)=18 _ (2)
xy-4x-2y+8=18
xy-2y-4x=10
y(x-2)=10+4x
y=(10+4x)/ (x-2)
put y=(10+4x)/(x-2) into 1
(10x+4x²)/(x-2)=A
dA/DX=(x-2)(10+8x)-(10x+4x²)/(x-2)²=0
10x+8x²-20-16x-10x-4x²=0
4x²-16x-20=0
x²-4x-5=0
x=5,x=-1
put x=5 into 2
3y-12=18
3y=30
y=10.