Respuesta :
Answer:
For least material to be used lengths of square base and sides = 10 units.
Step-by-step explanation:
Let the lengths of the square base and the sides = x feet, x feet and y feet
Area of the square base = x² feet
Volume of the rectangular prism = Area of the square base × Height
= x²y cubic feet
1000 = x²y
y = [tex]\frac{1000}{x^2}[/tex] -------(1)
Material used in the prism = Surface area of the rectangular prism
= 2(lb + bh + hl)
Here, h = height of the prism
l = length of the base
w = Width of the base
Material to be used (S) = 2(xy + x² + xy) - Area of lid
S = 2(x² + 2xy) - x²
S = x² + 2xy
Now by substituting the value of y from equation (1),
S = x² + [tex]2x(\frac{1000}{x^{2} })[/tex]
= x² + [tex]\frac{2000}{x}[/tex]
For least amount of material used,
We will find the derivative of the given function and equate it to zero.
S' = 2x - [tex]\frac{2000}{x^{2} }[/tex]
2x - [tex]\frac{2000}{x^{2} }[/tex] = 0
2x³ = 2000
x³ = 1000
x = 10 feet
From equation (1),
y = [tex]\frac{1000}{(10)^2}[/tex]
y = 10 feet
Therefore, for least amount of the material used lengths of square base and sides will be 10 feet.
