Respuesta :
Answer:
Maximum volume of the box will be 7.41 cubic feet.
Step-by-step explanation:
Open box has been made from a metal sheet measuring 3 ft and 8 ft.
Let four square pieces were removed from the four corners with one side measurement x ft.
Volume of the open box = Length × width × height
Length of the box = (3 - 2x) ft
Width of the box = (8 - 2x) ft
Height of the box = x ft
Volume of the box = (3 - 2x)(8 - 2x)x
V = (24 - 6x - 16x + 4x²)x
V = 24x - 22x² + 4x³
Now we take the derivative of V with respect to x and equate the derivative to zero,
[tex]\frac{d}{dx}(V)=\frac{d}{dx}(24x - 22x^{2}+4x^{3})[/tex]
V' = 24 - 44x + 12x²
V' = 0
12x² - 44x + 24 = 0
3x² - 11x + 6 = 0
3x² - 9x - 2x + 6 = 0
3x(x - 3) - 2(x - 3) = 0
(3x - 2)(x - 3) = 0
(3x - 2) = 0
and (x - 3) = 0
Therefore, x = 3, [tex]\frac{2}{3}[/tex]
For x = 0.67
Length of the box = (3 - 2x) = 3 - 1.34
= 1.66 ft
Width of the box = (8 - 2x) = 8 - 1.34
= 6.66 ft
Volume of the box = 0.67 × 1.66 × 6.66
V = 7.41 cubic feet.
Similarly, for x = 3,
Length of the box = (3 - 2\times 3) = -3
which is negative but the length of the box can not be negative.
Therefore, maximum volume of the box will be 7.41 cubic feet.
The maximum volume of the open box will be 7.41 ft³
How to calculate the volume?
From the length, width, and height of the formed box, the volume will be:
= Length × Width × Height
= (3 - 2x)(8 - 2x)(x)
= 4x³ - 22x² + 24x
In order to find the maximum volume of the box, we'll need to find the first derivative which will be 12x² - 44x + 24.
Therefore, this will be solved further below:
0 = 12x² - 44x + 24.
0 = 3x² - 9x - 2x + 6
3x(x - 3) - 2(x - 3) = 0
x - 3 = 0
x = 3
Also, 3x - 2 = 0, x = 2/3.
Therefore, the maximum length will be:
= Length × Width × Height
= [3 - 2(0.67)] + [8 - 2(0.67)] + 0.67
= 7.41 ft³
In conclusion, the correct option is 7.41ft³.
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