Applications-Of-Derivatives Question 431
Question: What is the area of the largest rectangular field which can be enclosed with 200 m of fencing?
Options:
A) $ 1600,m^{2} $
B) $ 2100,m^{2} $
C) $ 2400,m^{2} $
D) $ 2500,m^{2} $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Let length and breadth of rectangular field be x and y respectively
$ \therefore 2(x+y)=200\Rightarrow y=100-x $ and area, $ A=xy $ $ =x(100-x)\because \frac{dA}{dx}=100-2x $ Put $ \frac{dA}{dx}=0 $ for maxima or minima $ 100-2x=0 $
$ \Rightarrow x=50\Rightarrow y=50 $ Now, $ \frac{d^{2}A}{dx^{2}}=-2<0 $ , which shows maximum, independent of values of x and y, but only when they are equal.
$ \therefore $ A is maximum at $ x=50 $ . Hence, required area $ =50(100-50) $ $ =50\times 50=2500,m^{2} $
BETA
