Respuesta :
The dimensions of the given rectangular box are:
L = 15.874 cm , B = 15.874 cm , H = 7.8937 cm
Step-by-step explanation:
Let us assume that the dimension of the square base = S x S
Let us assume the height of the rectangular base = H
So, the total area of the open rectangular box
= Area of the base + 4 x ( Area of the adjacent faces)
= S x S + 4 ( S x H) = S² + 4 SH ..... (1)
Also, Area of the box = S x S x H = S²H
⇒ S²H = 2000
[tex]\implies H = \frac{2000}{S^2}[/tex]
Substituting the value of H in (1), we get:
[tex]A = S^2 + 4 SH = S^2 + 4 S(\frac{2000}{S^2}) = S^2 + (\frac{8000}{S})\\\implies A = S^2 + (\frac{8000}{S})[/tex]
Now, to minimize the area put :
[tex](\frac{dA}{dS} ) = 0 \implies 2S - \frac{8000}{S^2} = 0\\\implies S^3 = 4000\\\implies S = 15.874 \approx 16 cm[/tex]
Putting the value of S = 15.874 cm in the value of H , we get:
[tex]\implies H = \frac{2000}{S^2} = \frac{2000}{(15.874)^2} = 7.8937 cm[/tex]
Hence, the dimensions of the given rectangular box are:
L = 15.874 cm
B = 15.874 cm
H = 7.8937 cm
