Question

The highway department is planning to build a picnic area for motorist along a major highway. It is to be rectangular with an area of 5000 sq. m is to be fenced off on the three sides not adjacent to the highway. What is the least amount of fencing that will be needed to complete the job?
a. 500m
b. 300m
c. 200m
d. 400m

Answer 1

answer is 200 metres

Answer 2

let length=l and width=w
area =lw =5000 ----> l=5000/w

the fencing is minimum means that the side adjacent to the highway is a length of the rectangle (the bigger side)

amount of fencing = F(x) = 2w+l = 2w + 5000/w

F'(x) = 2 - 5000/(w^2)

fencing is minimum so F'(x)=0
2 - 5000/(w^2) =0
2= 5000/w^2
2w^2=5000
w=50
l=5000/w = 1000

amount of fencing=2w+l = 200

Answer 3

Only the positive value x 100 lies in the interval x 0. Since this is the only
critical value in the interval, you can apply the second derivative test for absolute
extrema. In particular, the second derivative is F(x) , which is positive when
x 0. Hence, the critical point at x 100 is the absolute minimum of F on the interval. That is, the least amount of fencing needed to complete the job is F(100) 200
yards

Answer 4

say, the adjacent side to highway is of length a and the others are b
then we have to find minimum of a+2b, while ab is given, i.e 2ab is given too
so, a+2b is minimum when a=2b
ab=5000
so, 2b^2=5000
so, b=50
a=100
a+2b=200 :)

Answer 5

thanks god bless 3 more to go