The dimension of the box will be (40 inch × 40 inch × 10 inch).
How to determine the maximum and minimum values of an expression using differentiation?
Any given expression can have both minimum and maximum outcomes for any number in a real world scenario, which can be determined using differentiation; first order differentiation when equated to zero reveals the value that will solute to produce the maximum or minimum and then the upon differentiating the obtained expression again and then substituting the variable in acceptable positions, if the outcome is a positive number then the obtained number in first differentiation suggests a minimum value of the given expression.
Given, the volume of the box described = 4000 inch³
Let the dimension of the base of the box be = 'a' inch × 'a' inch = a² inch²
Therefore, height of the box = (4000 inch³)/(a² inch²) = (4000/a²) inch
Total surface area of the described box = S = a² + [4a×(4000/a²)]
= [a² + 16000/a] inch
For the limiting values of S, dS/da = 0 ⇒ d[a² + 16000/a]/da = 0
⇒ 2a - [16000/a²] = 0 ⇒ 2a = 16000/a² ⇒ a³ = 8000 ⇒ a = ∛8000 = 20
Therefore, the value of a is 20 inch.
Thus, height of the box = 4000/a² = 4000/400 = 10 inch
For S to have the minimum limited value, d²S/da² > 0 for a = 20 inch, which is satisfied by the available value of S.
∴ The minimal surface area = a² + 16000/a = 400 + (16000/400)
= 440 inch²
We are aware that the least amount of material will be needed to build the box when its overall surface area is the smallest.
Thus, to minimize the amount of material in consideration, the dimension of the box will be (40 inch × 40 inch × 10 inch).
To learn more about minimization of materials, tap on the link below:
https://brainly.com/question/15522368
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