## IB Math Studies Optimization problems

Number and algebra, Descriptive statistics, Logic, sets and probability, Statistical applications, Geometry and trigonometry, Mathematical models, Introduction to differential calculus.

### IB Math Studies Optimization problems

IB Mathematical Studies SL – Calculus, Derivatives, Differentiation, Optimization problems

How can we find the largest area of a rectangular region having perimeter 600 m ?

Thanks
hill

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Joined: Tue Mar 19, 2013 8:49 pm

### Re: IB Math Studies Optimization problems

IB Mathematical Studies SL – Calculus, Derivatives, Differentiation, Optimization problems

Let $x \and\ y$ be the dimensions of the rectangular region and $A$ be its area. We want to find the largest value of A.
We know that $A=xy$. We need to eliminate $y$ using the fact that $2x +2y=600\Rightarrow x+y=300 \Rightarrow y=300-x$

Thus the function described the area can be written as
$A(x)=x(300-x)=-x^2+300x$

Therefore the problem now is to maximize the function $A(x)$
To find the maximum of $A(x)$ , we find the stationary points

$A'(x)= -2x+300=0 \Rightarrow x=150$

Since $A'(x)= -2x+300>0 \Rightarrow x<150$

the function $A(x)$ is increasing for $x<150$

and

$A'(x)= -2x+300<0 \Rightarrow x>150$

the function $A(x)$ is decreasing for $x>150$

So, the function $A(x)$ has a maximum when $x=150$

and the largest area of a rectangular region is

$A(100)= -150^2+300(150)=-22500+45000=22500 m^2$

Therefore the rectangular is a square $(x=y=150 m)$

Hope these help!!
elizabeth

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Joined: Mon Jan 28, 2013 8:09 pm

### Re: IB Math Studies Optimization problems

Thanks Elisabeth!!
hill

Posts: 0
Joined: Tue Mar 19, 2013 8:49 pm