Question

a rectangular piece of cardboard has a length ( l = 18 - 2w ) and width ( w ). an open box is to be constructed by cutting 5 inch squares from each corner. what is the maximum volume of the box created in this fashion? give your answer rounded to 2 decimal places. do not do any intermediate rounding. remember, the formula for the volume of a rectangular prism is ( v = lwh ).

Explanation:

Step1: Express the volume function

The length of the box is \(L = 18-2w\), the width of the box is \(w - 10\) (since we cut 5 - inch squares from each side of the width), and the height \(H=5\).
Using the volume formula \(V = LWH\), we substitute the values:
\(V(w)=5(18 - 2w)(w - 10)\)
Expand the expression:

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Step2: Find the derivative of the volume function

The derivative of \(V(w)\) using the power rule \((x^{n})^\prime=nx^{n - 1}\) is \(V^\prime(w)=-20w + 190\)

Step3: Set the derivative equal to zero and solve for \(w\)

Set \(V^\prime(w)=0\), so \(-20w+190 = 0\)
\(20w=190\), then \(w=\frac{190}{20}=9.5\)

Step4: Calculate the volume

Substitute \(w = 9.5\) into the volume function \(V(w)\)
\(V(9.5)=-10\times(9.5)^{2}+190\times9.5 - 900\)

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Answer:

\(102.50\)