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.