Question

Outside of the United States, the common paper size is called a4 and measures 21 by 29.7 centimeters. Let V(x) = (21 - 2x)(29.7 - 2x)(x) be the volume in cubic centimeters of a box made from A4 paper by cutting out squares of side length x in centimeters from each corner and then folding up the sides.

What is a reasonable domain domain for V in this context? Explain of show your reasoning

Answer 1

The domain, should be from x = 0 cm to x = 10.5 cm

Explanation:

The given parameters are;

The size of the A4 paper = 21 centimeters × 29.7 centimeters

The function for the volume V(x) = (21 - 2·x)·(29.7 - 2·x)·(x)

Therefore, we have;

V(x) = (21 - 2·x)·(29.7 - 2·x)·(x) = 4·x³ - 101.4·x² + 623.7·x

Whereby the volume becomes so small, we have;

(21 - 2·x)·(29.7 - 2·x)·(x) = 0 cm³

x = 0 cm. or x = 21/2 = 10.5 cm. or x = 29.7/2 = 14.85 cm.

However, by differentiation, to find the minimum point, we have;

d(4·x³ - 101.4·x² + 623.7·x)/dx = 0 = 12·(x² - 16.9·x + 51.975)

Factorizing online, gives (x - 4.04234)(x - 12.8577)

The minimum value occurs at x = 12.8577

The maximum value occurs at x = 4.04234

The minimum volume becomes; V(12.8577) = (21 - 2·12.8577)·(29.7 - 2·12.8577)·(12.8577) = -241.58 cm³

The maximum volume becomes; V(4.04234) = (21 - 2·4.04234)·(29.7 - 2·4.04234)·(4.04234) ≈ 1128.5 cm³

Therefore, the acceptable range should be from where the volume is the minimum real volume to the maximum real volume, which is from 0 cm³ to approximately 1128.5 cm³

The domain, therefore, should be from x = 0 cm to x = 10.5 cm.