A rectangular box, with a square base and open at the top is to be constructed. If the volume of the box needs to be 108 cubic feet, what is the minimum surface area?
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To find the minimum surface area of the box, we need to determine the dimensions of the box.
Let's denote the side length of the square base as 's' and the height of the box as 'h'. Since the box is open at the top, the surface area of the box consists of the four sides of the square base and the height of the box.
From the given information, we know that the volume of the box is 108 cubic feet. The volume of a rectangular box is given by the formula V = l * w * h, where 'l' represents the length, 'w' represents the width, and 'h' represents the height. Since the base of the box is square, the length and width are the same and equal to 's'. Therefore, we can rewrite the formula as V = s * s * h = 108.
We want to find the minimum surface area while keeping the volume constant. The surface area of the box is given by the formula A = 4s^2 + sh, where 'A' represents the surface area.
To find the minimum surface area, we need to express the surface area formula in terms of a single variable (either 's' or 'h') and then find the critical points by taking the derivative and setting it equal to zero.
We can use the equation V = s^2 * h = 108 to express 'h' in terms of 's'. Solving for 'h', we get h = 108 / s^2.
Substituting this value of 'h' into the surface area formula, we get A = 4s^2 + s * (108 / s^2) = 4s^2 + 108 / s.
Now, we can find the derivative of A with respect to 's' and set it equal to zero to find the critical points.
dA/ds = 8s - 108 / s^2 = 0
To simplify the equation, we can multiply both sides by s^2 to get:
8s^3 - 108 = 0
Simplifying further, we have:
s^3 - 13.5 = 0
To solve for 's', we can take the cube root of both sides:
s = ∛(13.5) ≈ 2.63 ft (rounded to two decimal places)
Now that we have the value of 's', we can substitute it back into the equation for 'h' to find the value of 'h':
h = 108 / s^2 ≈ 9.18 ft (rounded to two decimal places)
Therefore, the minimum surface area of the box is given by:
A = 4s^2 + sh = 4(2.63)^2 + 2.63(9.18) ≈ 102.43 square feet (rounded to two decimal places).
To find the minimum surface area of the rectangular box, we need to optimize the dimensions of the box to minimize the surface area.
Let's assume that the side length of the square base is "x" and the height of the box is "h".
The volume of a rectangular box is given by:
Volume = Length × Width × Height
In this case, the length and width of the base are the same (since it is a square). So we have:
Volume = x × x × h
Since the volume needs to be 108 cubic feet, we can write the following equation:
x^2 * h = 108
Now, let's express the height "h" in terms of "x" and substitute it back into the equation to have a single variable equation:
h = 108 / (x^2)
To minimize the surface area, we need to minimize the sum of all the sides' areas.
The surface area of the rectangular box is given by:
Surface Area = 2lw + lh + lh
= 2(x * x) + x * h + x * h
= 2x^2 + 2xh
= 2x^2 + 2x(108 / x^2) (substituting the value of h)
= 2x^2 + 216 / x
To find the minimum surface area, we need to find the critical points by taking the derivative of the surface area equation and setting it equal to zero:
d(Surface Area) / dx = 0
Differentiating the equation, we get:
d(Surface Area) / dx = 4x - 216 / x^2
Setting this equal to zero, we have:
4x - 216 / x^2 = 0
To solve for x, we can multiply the equation by x^2 to eliminate the denominators:
4x^3 - 216 = 0
Adding 216 to both sides:
4x^3 = 216
Dividing both sides by 4:
x^3 = 54
Taking the cube root of both sides:
x = ∛(54)
Approximating the cube root of 54:
x ≈ 3.78
Now that we have the value of x, we can substitute it back into the height equation to find "h":
h = 108 / (x^2)
h ≈ 108 / (3.78^2)
h ≈ 8.84
So, the dimensions of the box that minimize the surface area are approximately:
Side length of square base: x ≈ 3.78 ft
Height of the box: h ≈ 8.84 ft
Finally, we can calculate the minimum surface area by substituting these values into the surface area equation:
Surface Area ≈ 2(3.78^2) + 2(3.78)(8.84)
Surface Area ≈ 110.66 square feet
Therefore, the minimum surface area of the rectangular box is approximately 110.66 square feet.