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: Subtract the volume of Box 1 from Box 2

$$(4x^{5}-x^{4})-3x^{5}=x^{5}-x^{4}$$

Step2: Factor the result

$$x^{5}-x^{4}=x^{4}(x - 1)$$

Step3: Analyze the sign of the expression when \(x>1\)

Since \(x>1\), then \(x-1>0\) and \(x^{4}>0\) (for \(x
eq0\), and when \(x > 1\), \(x
eq0\)). So \(x^{4}(x - 1)>0\)

Answer:

Box 2 will hold more cereal. When \(x>1\), \(4x^{5}-x^{4}-3x^{5}=x^{4}(x - 1)>0\), which means \(4x^{5}-x^{4}>3x^{5}\)