Question

A box with a square base and open top must have a volume of 13,500 cm3. Find the dimensions of the box that minimize the amount of material used. sides of base cm height cm

Answer 1

Answer:

base: 30 cm

height: 15cm

Step-by-step explanation:

Let 'b' be the length of the side of the base, and 'h' the height of the box. The volume is given by:

$V=13,500=b^2*h$

The total area of material used is:

$A=b^2+4*(b*h)$

Using both equations to solve for 'b':

$h=\frac{13,500}{b^2}\\A=b^2+4*(b*\frac{13,500}{b^2})\\A=b^2+\frac{54,000}{b}\\\frac{dA}{db}=0=2b+\frac{54,000}{b^2}\\2b^3=54,000\\b=30\ cm$

The value of 'b' for which the derivate of the area function (A) equals zero, is the value that yields the minimum area.

If b=30, the height 'h' is:

$13,500=30^2*h\\h = 15\ cm$

Answer 2

Base of 30 cm by 30 cm and height of 15 cm

Let each side of the base measure x cm, and let the height of the base be h cm. Since the box has an open top, the surface area (A) of the base is:

A = x² + xh + xh + xh + xh

A = x² + 4xh     (1)

The volume (V) of the box is given by:

V = x * x * h = x²h

13500 = x²h

h = 13500/x²      (2)

Substitute h = 13500/x² in equation 1, this gives:

A = x² + 4x(13500/x²)

A = x²  + 54000/x

For minimal amount of material, dA/dx = 0, hence:

dA/dx = 2x - 54000/x²

2x - 54000/x² = 0

54000/x² = 2x

2x³ = 54000

x³ = 27000

x = ∛27000 = 30 cm

Therefore the square base is 30 cm by 30 cm.

h = 13500/x² = 13500/30² = 15 cm

The box has a base of 30 cm by 30 cm and height of 15 cm

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