A box with an open top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the box has the least possible surface area, find its dimensions.
Length of base=
Height=
L=2
H=1
To find the dimensions of the box with the least possible surface area, we need to minimize the surface area while maintaining the volume of 4 cubic meters.
Let's assume the length of the base of the square bottom is "x" meters.
Since the box has vertical sides, the height of the box will also be "x" meters.
The volume of the box is given as 4 cubic meters, so we have:
Volume = Length x Width x Height
4 = x^2 * x
4 = x^3
Take the cube root of both sides to solve for x:
∛4 = ∛(x^3)
2 = x
Therefore, the length of the base of the square bottom is 2 meters, and the height of the box is also 2 meters.
To find the dimensions of the box with the least possible surface area, we need to determine the length of the base and the height.
Let's assume the length of the base is "x" meters. Since the base is a square, the width and length of the base will both be "x" meters.
Given that the volume of the box is 4 cubic meters, we can set up the equation:
Volume of the box = length × width × height
4 = x × x × height
4 = x^2 × height
Since we want to minimize the surface area, we need to minimize the amount of material used, which means minimizing the surface area.
The surface area of the box can be calculated by adding the areas of the bottom, top, front, back, left, and right sides:
Surface area = 2(Length × Width) + Length × Height + Width × Height
Since the bottom and top have the same area (Length × Width), we can simplify the surface area equation as:
Surface area = 2(Length × Width) + 2(Length × Height)
Substituting the values "x" for both Length and Width, we have:
Surface area = 2(x × x) + 2(x × height)
Surface area = 2x^2 + 2xheight
Now, we can express the height in terms of x by rearranging the equation we obtained from the volume:
4 = x^2 × height
height = 4/x^2
Substituting this expression for height in the surface area equation, we get:
Surface area = 2x^2 + 2x × (4/x^2)
Surface area = 2x^2 + 8/x
To minimize the surface area, we can take the derivative of the surface area equation with respect to x, set it equal to zero, and solve for x:
d(Surface area)/dx = 4x - 8/x^2 = 0
4x - 8/x^2 = 0
4x^3 - 8 = 0
x^3 = 2
x = ∛2
Now, we can substitute this value of x back into the expression for the height:
height = 4/x^2
height = 4/(∛2)^2
height = 4/∛4
height = (4/∛4) * (∛1/∛1)
height = (4/∛4) * (∛1/(∛1 * ∛4))
height = (4/∛4) * (∛4/(∛4 * ∛4))
height = (4/∛4) * (∛4/∛16)
height = (4/∛4) * (∛4/2)
height = 4/2
height = 2
Therefore, the length of the base (which is also the width) is approximately ∛2 meters, and the height is 2 meters.