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 cm^3 and 900 cm^3?

Question

I know the area "A" would equal (x+3)(x+5)(x+11) and would be
300<(x+3)(x+5)(x+11)<900, however, I have been having trouble solving the inequality. Is there another way to do this question or am I missing something? Step by step process would be greatly appreciated.

Answer 1

The inequality can be solved by factoring the polynomial.

300 < (x+3)(x+5)(x+11) < 900

300 < x^2 + 14x + 33 < 900

0 < x^2 + 14x - 267 < 600

0 < (x + 13)(x - 20) < 600

Therefore, x must be between -13 and 20.

Answer 2

AAAaannndd the bot gets it wrong yet again!
how did you wind up with a quadratic?

1.08 <= x <= 3.85

Answer 3

Thank you! I know the answers are 1.08 and 3.85 by checking desmos. However, the issue is showing my work algebraically to get that answer.

To answer your question I know the area of the box = length*width*height
and if each dimension increases by "x" it would lead to the quadratic I determined above.