Question

A rectangular box with a square base, an open top, and a volume of 32,000 cm3 is to be made. What is the minimum surface area for the box

Answer 1

the minimum surface area for the box is 4800 cm².

Forming the Equation of SurfaceArea

It is given that the given rectangular box is square-based and top is open. hence, it consists of  square and 4 rectangles.

Let the side of the square be a, and height of the box be h.

Then, the total surface area of the box will be given by,

S = a² + 4ah _________ (1)

Also, it is given that the volume of the box is, V = 32000 cm³

The volume of the rectangular box, V = a² h

Eliminating One of the Variables From the Equation

The volume of the rectangular box, V = a² h

⇒ a² h = 32000

⇒ h = 32000/a² _______ (2)

Substituting this value of h in equation (1), we get,

S = a² +4a(32000/a²)

S = a²+128000/a

Minimizing the Surface Area Equation

To find the minima, put, dS/da = 0

dS/da = 2a-128000/a²

⇒ 2a-128000/a² = 0

Multiplying the whole equation with a², we get,

2a³-128000 = 0

⇒ 2a³ = 128000

⇒ a³ = 128000/2

⇒ a³ = 64000

⇒ a = 40 cm

Calculating the Minimum Surface Area

From, equation (2), h = 32000/(40)²

h = 32000/1600

h = 20 cm

Now, substituting the computed values of a and h in equation (1), we get,

S = (40)² +4(40)(20)

S = 1600 +3200

S = 4800 cm²

∴ The minimum surface area of the box is 4800 cm².

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