Answer:
x = 5 inches
Step-by-step explanation:
The lateral surface area (LSA) of a three-dimensional object includes the surface area of its sides, but excludes the area of the bases or ends.
To find the length of the box (x), we first need to determine the lateral surface area (LSA) of the box, considering that the white label covers 60% of the LSA.
The total surface area of the label is:
[tex]\textsf{SA of label} = 2(3 \cdot 3 + 3 \cdot 2)\\\\\textsf{SA of label} = 2(9 + 6)\\\\\textsf{SA of label} = 2(15)\\\\\textsf{SA of label} = 30 \; \sf in^2[/tex]
Given that the label is 60% of the LSA then:
[tex]\textsf{SA of label} =0.6 \cdot \textsf{LSA}\\\\30 = 0.6\cdot\textsf{LSA}\\\\\textsf{LSA} = \dfrac{30}{0.6}\\\\\textsf{LSA} = 50 \; \sf in^2[/tex]
Therefore, the LSA of the box is 50 in².
Given that the white label covers 60% of the lateral surface area, the bases of the box must be the two sides (ends) with dimensions 2 in × 3 in, as these are the sides that the label doesn't cover. Therefore, the expression for the lateral surface area of the box is:
[tex]\textsf{LSA} = 2(3x + 2x)\\\\\textsf{LSA} = 2(5x)\\\\\textsf{LSA} = 10x[/tex]
To find the length of the box (x), substitute the total LSA (50 in²) into the equation for the LSA and solve for x:
[tex]10x = 50\\\\\dfrac{10x}{10}=\dfrac{50}{10}\\\\x = 5[/tex]
Therefore, the length of the box is:
[tex]\LARGE\boxed{\boxed{x = 5\; \sf inches}}[/tex]
