Answer:
x = 12 m and y = 22 m
Step-by-step explanation:
Total area = 264 [tex]m^2[/tex]
∴ xy = 264
[tex]$y=\frac{264}{x}$[/tex] ............(1)
Cost function = [tex]C(x,y) = 2 x (15.60) + 2y(15.60) + 2x(13)[/tex]
[tex]C(x,y) = 57.2 x + 31.2y[/tex]
Therefore, using (1),
[tex]$C(x) = 57.2x+31.2 \left(\frac{264}{x} \right)$[/tex]
[tex]$C(x) = 57.2x+\frac{8236.8}{x} \right)$[/tex]
So, cost C(x) minimum where C'(x) = 0
[tex]$C'(x) = 57.2 - \frac{8236.8}{x^2}=0$[/tex]
[tex]$x^2=\frac{8236.8}{57.2}$[/tex]
[tex]$x^2=144$[/tex]
[tex]$x=12$[/tex] m
Therefore, [tex]$y=\frac{264}{x}$[/tex]
[tex]$=\frac{264}{12}$[/tex]
= 22 m
So the dimensions are x = 12 m and y = 22 m.
