what are the dimensions of the lightest rectangular box without an open top whose base is a square and whose height is two less than the length of the side of the base?
To find the dimensions of the lightest rectangular box without an open top, we'll start by defining the variables:
Let's denote:
- Length of the side of the square base = L
- Height of the rectangular box = H
From the given information, we know that the height of the box is two less than the length of the base's side. Therefore, we have:
H = L - 2
The weight of an object is directly proportional to its volume. In the case of a rectangular box, the volume is given by the formula V = L * W * H, where L is the length, W is the width, and H is the height.
Since we want to find the lightest box, we need to minimize the volume, V. Since the base is a square, the length and width are the same, so we can write the volume as V = L * L * (L - 2).
Now, to find the dimensions of the lightest box, we need to find the values of L and H that minimize the volume V.
To do this, we'll differentiate the volume function V = L * L * (L - 2) with respect to L, set the derivative equal to zero, and solve for L:
dV/dL = 0
Taking the derivative and solving for L:
dV/dL = 2L^2 - 4L + 2 = 0
Dividing both sides by 2:
L^2 - 2L + 1 = 0
Factoring the equation:
(L - 1)^2 = 0
Solving for L:
L - 1 = 0
L = 1
Now that we have the value of L, we can find the height H using the equation H = L - 2:
H = 1 - 2
H = -1
However, since the height of a box cannot be negative, we can conclude that there is no solution that satisfies the given conditions.
To find the dimensions of the lightest rectangular box without an open top, we'll break down the problem into steps:
Step 1: Define the variables.
Let's assume:
- Side length of the base of the box = "s"
- Height of the box = (s - 2)
Step 2: Determine the dimensions of the box.
Since the base is a square, all sides are equal. Therefore, the dimensions of the base will be (s x s).
For the box without an open top, it will have six sides. Four sides will have the dimensions (s x (s - 2)), and the other two sides will be (s x s).
Step 3: Calculate the weight.
To find the lightest box, we need to consider the weight of each side. Let's assume the thickness of the material used to build the box is negligible, making all sides uniformly thin.
The weight of each side is proportional to its surface area. The surface area of each side of the box can be calculated using the formula: Length x Width.
The weight of the box will be the sum of the surface areas of all six sides.
Step 4: Simplify the weight equation.
The weight equation for the box is:
(4 x (s x (s - 2))) + (2 x (s x s))
Step 5: Optimize the weight.
To find the dimensions of the lightest box, we need to minimize the weight equation. We can achieve this by finding the value of "s" that minimizes the equation.
Step 6: Solve for "s".
Expand the weight equation and simplify it:
(4s² - 8s) + (2s²)
6s² - 8s
To minimize the equation, we can take the derivative and set it equal to zero:
12s - 8 = 0
12s = 8
s = 8/12
s = 2/3
Therefore, the value of "s" is 2/3.
Step 7: Find the dimensions.
Substitute the value of "s" back into the dimensions:
- Base dimensions: (2/3) x (2/3)
- Height: (2/3 - 2)
So, the dimensions of the lightest rectangular box without an open top, given the conditions, would be approximately (2/3 x 2/3 x (-4/3)).