Question

suppose celine wants to choose a box that maximizes the amount of cereal it can hold.

volume of box 1: $3x^5$
volume of box 2: $4x^5 - x^4$

if celine decides the width of the cereal boxes will be greater than 1, which box will hold more cereal? explain.

Explanation:

Step1: Set up volume comparison

We compare the volumes by subtracting Box 1's volume from Box 2's volume:
$$(4x^5 - x^4) - 3x^5$$

Step2: Simplify the expression

Combine like terms to simplify the difference:
$$4x^5 - 3x^5 - x^4 = x^5 - x^4 = x^4(x - 1)$$

Step3: Analyze the result for $x>1$

Since $x>1$, $x^4>0$ and $x-1>0$. The product of two positive values is positive:
$$x^4(x - 1) > 0$$
This means Box 2's volume minus Box 1's volume is positive, so Box 2 has a larger volume.

Answer:

Box 2 will hold more cereal. When $x>1$, the difference between Box 2's volume and Box 1's volume is $x^4(x-1)$, which is positive (as both $x^4$ and $x-1$ are positive for $x>1$), so $4x^5 - x^4 > 3x^5$.