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. which did you include in your response? if a box has a greater volume, it can hold more cereal. box 2 can hold more cereal. although the value of x is unknown, it represents a length, so it must be greater than zero. for any $x>1$, the volume of box 2 is greater than the volume of box 1.

Explanation:

Step1: Find the difference in volumes

Let \(V_1 = 3x^{5}\) and \(V_2=4x^{5}-x^{4}\). Calculate \(V_2 - V_1=(4x^{5}-x^{4})-3x^{5}\).

$$
LATEXBLOCK0
$$

Step2: Analyze the sign of the difference

Since \(x>1\), then \(x-1>0\) and \(x^{4}>0\) (because for any real \(x
eq0\), \(x^{4}>0\), and here \(x > 1\)). So \(V_2 - V_1=x^{4}(x - 1)>0\).

Answer:

Box 2 will hold more cereal. Since \(x>1\), the difference \(V_2 - V_1=x^{4}(x - 1)>0\), which means \(V_2>V_1\).