Answer:
(a) From the graph we get that the points (100, 30 000) and (200, 30 000) are on the parabola, therefore the largest possible value of x such that the are is 30 000 m² is
[tex]200\text{ m.}[/tex]
(b) To find the largest possible area we calculate the vertex of the parabola.
Taking the given equation to its standard form we get:
[tex]\begin{gathered} A=-1.5(x^2-300x)=-1.5((x-150)^2-150^2), \\ A=-1.5(x-150)^2+33\text{ 750.} \end{gathered}[/tex]
Therefore, the coordinates of the vertex are:
[tex](150,33750)\text{.}[/tex]
From the vertex we get that, the value of x that maximizes the area is x=150.
Substituting x=150 in 3x+2y=900, and solving for y, we get:
[tex]\begin{gathered} 450+2y=900, \\ 2y=450, \\ y=225. \end{gathered}[/tex]
The values of x and y that maximize the area are x=150m, y=225 m.
(c) From the previous step we get that the maximum area of the paddock is 33 750 m².
