Question

compare the area of the bases. if a base has a greater area, it will take up more space on the shelf.

base area of box 1: $3x^2$
base area of box 2: $4x^2 - x$

which box is likely to occupy more space on the shelf? explain. try substituting different values for $x$.

Explanation:

Step1: Set up area difference

Find the difference between Box 2's area and Box 1's area to compare them.
$\text{Difference} = (4x^2 - x) - 3x^2$

Step2: Simplify the difference

Combine like terms to simplify the expression.
$\text{Difference} = 4x^2 - 3x^2 - x = x^2 - x = x(x - 1)$

Step3: Test positive x values (since x represents a length, $x>0$)

Case 1: $0 < x < 1$ (e.g., $x=0.5$)

$\text{Difference} = 0.5(0.5 - 1) = 0.5(-0.5) = -0.25$
Box 1 area: $3(0.5)^2 = 0.75$; Box 2 area: $4(0.5)^2 - 0.5 = 1 - 0.5 = 0.5$

Case 2: $x = 1$

$\text{Difference} = 1(1 - 1) = 0$
Box 1 area: $3(1)^2 = 3$; Box 2 area: $4(1)^2 - 1 = 3$

Case 3: $x > 1$ (e.g., $x=2$)

$\text{Difference} = 2(2 - 1) = 2(1) = 2$
Box 1 area: $3(2)^2 = 12$; Box 2 area: $4(2)^2 - 2 = 16 - 2 = 14$

Answer:

When $0 < x < 1$, Box 1 occupies more space on the shelf. When $x = 1$, both boxes occupy the same amount of space. When $x > 1$, Box 2 occupies more space on the shelf. Since $x$ represents a length of a box, $x$ is a positive real number; for practical box sizes where $x > 1$ (typical positive lengths greater than 1 unit), Box 2 will take up more space.