Question

Q15. Consider a box with dimensions 3 cm × 5 cm × 11 cm. If all of its dimensions are increased by x cm, what values of x will give a box with a volume between 300 cm3 and 900 cm3?

Answer 1

To find the volume of the box before increasing its dimensions, we multiply its length, width, and height:
Volume before = 3 cm × 5 cm × 11 cm = 165 cm³

Now, let's find the volume after increasing its dimensions by x cm:
Length after = 3 cm + x cm
Width after = 5 cm + x cm
Height after = 11 cm + x cm

Volume after = (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm)

Since we want the volume to be between 300 cm³ and 900 cm³:
300 cm³ < (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm) < 900 cm³

Now, we can solve this inequality:
300 cm³ < (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm) < 900 cm³

Expanding the factors:
300 cm³ < (15 cm² + 8 cmx + x²) × (11 cm + x cm) < 900 cm³

Multiplying the binomials:
300 cm³ < 165 cm² + 91 cmx + 11 x² + 8 cmx + 8x² + x³ < 900 cm³

Combining like terms:
300 cm³ < 165 cm² + 99 cmx + 19 x² + x³ < 900 cm³

Now, we can factor and solve the inequality by considering each part:

The volume must be greater than 300 cm³:
165 cm² + 99 cmx + 19 x² + x³ - 300 cm³ > 0

The volume must be less than 900 cm³:
165 cm² + 99 cmx + 19 x² + x³ - 900 cm³ < 0

To find the values of x, we can use a graphing calculator or factoring techniques. The solutions for both parts of the inequality will give us the range of x values that will give a box with a volume between 300 cm³ and 900 cm³.