Answer:
The inequality that the shortest length must satisfy is [tex]\( x > 3 \).[/tex] This means that the shortest side length must be greater than 3 cm for the surface area to be larger than 144 cm².
Step-by-step explanation:
Let's denote the lengths of the sides of the box as [tex]\( x \), \( 2x \), and \( 2x \) (since the ratio of the side lengths is 1:2:2).[/tex]
The surface area of the box can be calculated using the formula for the surface area of a rectangular box:
[tex]\[ \text{Surface Area} = 2lw + 2lh + 2wh \][/tex]
Given that the lengths of the sides are[tex]\( x \), \( 2x \), and \( 2x \)[/tex], we can substitute these values into the formula:
[tex]\[ \text{Surface Area} = 2(x)(2x) + 2(x)(2x) + 2(2x)(2x) \][/tex]
[tex]\[ \text{Surface Area} = 4x^2 + 4x^2 + 8x^2 \][/tex]
[tex]\[ \text{Surface Area} = 16x^2 \][/tex]
Now, we want the surface area to be larger than 144 cm²:
[tex]\[ 16x^2 > 144 \][/tex]
Divide both sides by 16:
[tex]\[ x^2 > \frac{144}{16} \][/tex]
[tex]\[ x^2 > 9 \][/tex]
Take the square root of both sides (note that [tex]\( x \)[/tex] must be positive):
[tex]\[ x > 3 \][/tex]
So, the inequality that the shortest length must satisfy is [tex]\( x > 3 \).[/tex] This means that the shortest side length must be greater than 3 cm for the surface area to be larger than 144 cm².
