A box with a square base is to be constructed with a surface area of 726 square centimeters.
1. Draw a diagram of the box. Label the diagram appropriately with variables.
2. Write an objective equation and a constraint equation (label each one as objective or constraint).
3. Determine the dimensions of the box with maximum volume. Be sure to show you have in fact
found a maximum. Do not forget units in your final answer.
assuming an open box, then
x^2 + 4xz = 426
v = x^2 z = x^2 (426-x^2)/4x
= 426x - x^3
To solve this problem, let's go through each step:
1. Drawing a diagram:
Let's assume the base of the box has sides of length 's' centimeters, and the height of the box is 'h' centimeters. We can draw a rectangle with sides of length 's' and 'h' (representing the side walls of the box), and a square with sides of length 's' (representing the square base of the box).
2. Writing objective and constraint equations:
Objective equation: We want to find the dimensions that maximize the volume of the box. The volume of a rectangular box is given by V = lwh, where l, w, and h are the length, width, and height respectively.
In our case, since the base of the box is square, the length and width are the same, and we'll denote them as 's'. So, the volume equation becomes V = s^2h.
Constraint equation: The total surface area of the box is given as 726 square centimeters. The surface area of a rectangular box is calculated by summing the areas of all six faces. For our box, the surface area equation becomes A = 2s^2 + 4sh.
3. Determining the dimensions of the box with maximum volume:
To find the dimensions of the box that maximize the volume, we need to find the values of 's' and 'h' that satisfy the constraint equation and make the volume equation as large as possible.
To do this, we can substitute the value of A into the volume equation and simplify it:
V = s^2h = (726 - 4sh)h [Substituting A = 726 in the constraint equation]
V = 726h - 4sh^2 [Rearranging terms]
Now, we need to find the critical points of V in terms of h. To find these points, we'll take the derivative of V with respect to h and set it equal to zero:
dV/dh = 726 - 8sh [Taking the derivative of V with respect to h]
Setting dV/dh = 0, we get:
726 - 8sh = 0
sh = 726/8
sh = 90.75
Since 's' represents a length, it must be positive. Therefore, 'h' must be positive as well, so we can discard the negative solution for sh.
Now, substitute the value of sh = 90.75 back into the constraint equation to solve for 's':
A = 2s^2 + 4sh
726 = 2s^2 + 4s(90.75)
726 = 2s^2 + 363s
2s^2 + 363s - 726 = 0
Solving this quadratic equation will give us the value(s) of 's'.
Once we have the values of 's' and 'h', we can calculate the volume V = s^2h, and compare it to determine the maximum volume.
Note: The solution involves solving a quadratic equation, so the final answer would have units in centimeters, as the given quantities are in centimeters.