Parabolas Application Q3

Question

A farmer decides to fence a new rectangular paddock with the greatest possible area, using an existing fence for one side and 40 metres of fencing to make the other three sides. Let the area of the paddock be defined as A m²

 

a) Write an equation using x and y to describe the area of the paddock

b) Write an equation using x and y to describe the length of the new fence.

c) Rearrange the equation so that y is the subject.

d) Substitute this value of y into the equation for the area

e) Using the intercept method find the coordinates of the turning point

f) Sketch the graph

g) Using the graph to find the maximum area of the paddock and its dimensions.

 

I really need helps. Its parabola application and needs working out.

Answer 1

10 m x 20m

max area = 200 m^2

A=xy

40=2x+y

y=40-2x

A=x(40-2x)=40x-2x^2

A'=40-4x=0

x=40/4= 10 m

y=40-2(10)=20 m

A(x) =-2x^2 +40x = -2(x^2 -20x+ 100)+ 200

is a downward opening parabola with vertex = maximum = (0,200)