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

compare the area of the bases. if a base has a greater area, it will take up more space on the shelf.\nbase area of box 1: $3x^{2}$\nbase area of box 2: $4x^{2}-x$\nwhich box is likely to occupy more space on the shelf? explain. try substituting different values for $x$.

compare the area of the bases. if a base has a greater area, it will take up more space on the shelf.\nbase area of box 1: $3x^{2}$\nbase area of box 2: $4x^{2}-x$\nwhich box is likely to occupy more space on the shelf? explain. try substituting different values for $x$.

Answer

Explanation:

Step1: Find the difference of the areas

Let (A_1 = 3x^{2}) and (A_2=4x^{2}-x). Calculate (A_2 - A_1=(4x^{2}-x)-3x^{2}=x^{2}-x=x(x - 1)).

Step2: Analyze the sign of the difference

Case 1: When (x<0), (x) is negative and (x - 1) is negative, so (x(x - 1)>0), then (A_2>A_1). Case 2: When (0<x<1), (x) is positive and (x - 1) is negative, so (x(x - 1)<0), then (A_2<A_1). Case 3: When (x = 0), (A_2 - A_1=0), then (A_2=A_1). Case 4: When (x = 1), (A_2 - A_1=0), then (A_2=A_1). Case 5: When (x>1), (x) is positive and (x - 1) is positive, so (x(x - 1)>0), then (A_2>A_1).

Answer:

If (x<0) or (x > 1), Box 2 occupies more space. If (0<x<1), Box 1 occupies more space. If (x = 0) or (x=1), both boxes occupy the same amount of space.