Question

A rectangular piece of sheet metal

Answer 1

Answer:

x = 2.354 in

Maximum volume = 228.162 in³

Step-by-step explanation:

Size of the metal sheet = (18 in × 12 in)

Let the height of the box = x in

Then length of the box = (18 - 2x) in

Width of the box = (12 - 2x) in  

Now volume of the rectangular box (V) = Length × Width × height

                                                                 = (18 - 2x) × (12 - 2x) × x

                                                                 = x[18(12 - 2x) - 2x(12 - 2x)]

                                                                 = x[216 - 36x - 24x + 4x²]

                                                                 = x[216 - 60x + 4x²]

                                                                 = (4x³ - 60x² + 216x) in³

For maximum volume, we will find the derivative of the volume and equate it to zero.

$\frac{dV}{dx}=\frac{d}{dx}(4x^3-60x^2+216x)$

V' = 12x² - 120x + 216

V' = 0

12x² - 120x + 216 = 0

x² - 10x + 18 = 0

By quadratic formula,

x = $\frac{10\pm\sqrt{(-10)^2-4(1)(18)}}{2(1)}$

x = 5 ± √7

x = 7.646, 2.354

But for 7.646, volume of the box will be negative.

Therefore, (x = 2.354 in) will be value of x for maximum volume of the box.

Maximum volume of the box = 4(2.354)³ - 60(2.354)² + 216(2.354)

                                                = 228.162 in³