. Find the dimensions of the rectangle with a perimeter of 100 m that has the
largest possible area.
ANSWER: 𝑥 = 𝑦 = 25 are the dimensions
that will gives the maximum area.
I WANT THE STEPS
ITS OPTIMIZATION
x = Length
y = Width
P = Perimeter
P = 2 x + 2 y
P = 2 ( x + y )
Perimeter = 100 m so:
100 = 2 ( x + y )
Divide both sides by 2
50 = x + y
x + y = 50
Subtract x to both sides
y = 50 - x
A = Area
A = x ∙ y
A = x ∙ ( 50 - x )
A = 50 x - x²
First derivative:
A'(x) = 50 - 2 x
The function has a maximum or minimum if first derivative = 0
In this case:
A'(x) = 0
50 - 2 x = 0
Add 2 x to both sides
50 = 2 x
2 x = 50
Divide both sides by 2
x = 25
x = 25 m
Second derivative test:
If f"(x) < 0 , then f has a maximum at x
If f"(x) > 0 , then f has a minimum at x
If f"(x) = 0 , the test is inconclusive
In this case:
A"(x) = ( A'(x) ) ' = ( 50 - 2 x )' = 0 - 2 = - 2
A"(x) < 0
So area has maximum at x = 25 m
y = 50 - 25 = 25
y = 25 m
Area has maximum at x = y = 25 m