Question: An open-top box is to be constructed from a sheet of tin that measures 24 inches by 12 inches by…

An open-top box is to be constructed from a sheet of tin that measures 24 inches by 12 inches by cutting out squares from each corner as shown and then folding up the sides. Let V(x) denote the volume of the resulting box.

[Diagram of a rectangle with dimensions 24 inches by 12 inches, and squares of side length x cut out from each corner.]

Step 2 of 2: Among the values of x for which V(x) = 0, which are physically possible?

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Solution

Step 1 Define the volume function: \[ V(x) = x(24 - 2x)(12 - 2x) \] Step 2 Set \( V(x) = 0 \): \[ x(24 - 2x)(12 - 2x) = 0 \] Step 3 Solve for \( x \): \[ x = 0 & \] \[ 24 - 2x = 0 \implies x = 12 & \] \[ 12 - 2x = 0 \implies x = 6 & \] Step 4 Determine physically possible solutions. Since the squares cut out must have positive dimensions and less than half the sheet’s smallest side: \[ 0 < x < 6 \] Therefore, the physically possible values of \( x \) are: \[ x = 0 \quad \text{and} \quad x = 6 \]