# The total surface area of a square-based open top rectangular box is 12

square units. Find the dimensions of the box such that the volume is the
maximum.

## 4 s h + s^2 = 12 so h = (12-s^2)/4s

v = s^2 h

v = s^2 (12-s^2)/4s = (12 s^2 -s^4)/4s

v = 3 s -(1/4) s^3

dv/ds = 0 for max = 3 - (3/4) s^2

(3/4) s^2 = 3

s^2 = 4
s = 2

h = (12 - 4)/8 = 1

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## To find the dimensions of the box such that the volume is the maximum, we need to maximize the volume function subject to the constraint of the total surface area.

Let's denote the sides of the square base as x, and the height of the box as h.

The volume of the rectangular box is given by V = x^2h.

The total surface area is given by SA = 2(x^2) + 4(xh).

According to the problem, the total surface area is 12 square units. So we have the equation:

2(x^2) + 4(xh) = 12.

Now, let's solve for h in terms of x:

2(x^2) + 4(xh) = 12
2x^2 + 4xh = 12
x^2 + 2xh = 6
2xh = 6 - x^2
h = (6 - x^2) / (2x).

Now substitute this expression for h into the volume function to get the volume as a function of x:

V = x^2h
V = x^2((6 - x^2) / (2x))
V = (x(6 - x^2)) / 2.

To find the dimensions of the box that maximize the volume, we need to find the value of x that maximizes the volume function V(x). To do this, we can take the derivative of V(x) with respect to x, set it equal to zero, and solve for x.

Let's differentiate V(x) with respect to x:

dV/dx = [(6 - x^2) / 2] - [(2x)(-2x)] / 2
dV/dx = (6 - x^2) / 2 + 2x^2 / 2
dV/dx = (6 - x^2 + 2x^2) / 2
dV/dx = (x^2 + 6) / 2.

Setting dV/dx equal to zero:

(x^2 + 6) / 2 = 0
x^2 + 6 = 0
x^2 = -6.

Since we cannot have a negative length for the side of a square, this means there is no real solution for x. Therefore, we cannot find dimensions of the box such that the volume is maximum.

Hence, there are no dimensions of the box that maximize the volume given the constraint of the total surface area.

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## To find the dimensions of the box that maximize its volume, we first need to understand the geometry of the box and its constraints.

Let's assume that the box has a square base with each side measuring "x" units. Since it is an open-top box, we also need to consider the height of the box, denoted by "h" units.

The total surface area of the box is given as 12 square units. The surface area of a rectangular box comprises four sides and a base. Since the base is a square, it has an area of x * x = x^2, and each of the four sides has an area of x * h.

Therefore, the total surface area can be expressed as:

Total Surface Area = x^2 + 4(x * h)

Since we are given that the total surface area is 12 square units, we can write the equation:

x^2 + 4(x * h) = 12

To find the maximum volume, we need to consider the volume formula for a rectangular box:

Volume = Length * Width * Height
= x * x * h
= x^2 * h

Now, let's solve the equation for the surface area to find an expression for "h" in terms of "x":

x^2 + 4(x * h) = 12
x^2 + 4xh = 12
4xh = 12 - x^2
h = (12 - x^2) / (4x)
h = 3 - (x^2 / 4x)
h = 3 - (x / 4)

Now, substitute this expression for "h" into the volume equation:

Volume = x^2 * (3 - (x / 4))
= 3x^2 - (x^3 / 4)

To maximize the volume, we need to find the critical points by taking the derivative of the volume function and setting it to zero.

d(Volume) / dx = 6x - (3x^2 / 4) = 0

Simplifying the equation:

24x - 3x^2 = 0
3x(8 - x) = 0

From this equation, we find two critical points: x = 0 and x = 8.

However, x = 0 does not make sense in the context of a physical box, as it would have zero volume.

Therefore, the only valid critical point is x = 8.

Now, substitute this value back into the equation for "h":

h = 3 - (x / 4)
h = 3 - (8 / 4)
h = 3 - 2
h = 1

So the dimensions of the box that maximize its volume are:

Length (x) = 8 units
Width (x) = 8 units
Height (h) = 1 unit