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?
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.
AAAaannndd the bot gets it wrong yet again!
how did you wind up with a quadratic?
1.08 <= x <= 3.85
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.
To find the values of x that give a box with a volume between 300 cm^3 and 900 cm^3, you can solve the inequality (x+3)(x+5)(x+11) > 300 and (x+3)(x+5)(x+11) < 900.
Let's start with (x+3)(x+5)(x+11) > 300:
1. Expand the expression: (x+3)(x+5)(x+11) = x^3 + 19x^2 + 113x + 165.
2. Set this expression greater than 300: x^3 + 19x^2 + 113x + 165 > 300.
3. Simplify by subtracting 300 from both sides: x^3 + 19x^2 + 113x - 135 > 0.
Now let's solve (x+3)(x+5)(x+11) < 900:
1. Expand the expression: (x+3)(x+5)(x+11) = x^3 + 19x^2 + 113x + 165.
2. Set this expression less than 900: x^3 + 19x^2 + 113x + 165 < 900.
3. Simplify by subtracting 900 from both sides: x^3 + 19x^2 + 113x - 735 < 0.
Now, to solve these inequalities, we can use a graphing calculator or computer software to graph the functions y1 = x^3 + 19x^2 + 113x - 135 and y2 = x^3 + 19x^2 + 113x - 735.
The values of x that satisfy the first inequality (x+3)(x+5)(x+11) > 300 will be the values for which y1 > 0.
The values of x that satisfy the second inequality (x+3)(x+5)(x+11) < 900 will be the values for which y2 < 0.
By analyzing the graph, you can determine the intervals of x values that satisfy each inequality.
Alternatively, you can use numerical methods such as Newton's method or the bisection method to find the solutions to these inequalities. These methods involve iterative calculations to get increasingly accurate estimates of the solutions.
Remember to check the endpoints of the obtained intervals to ensure that they satisfy the inequalities as well.
I hope this step-by-step process helps you solve the inequality and find the values of x that give a box with a volume between 300 cm^3 and 900 cm^3.
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.