An open-top box is to be made by cutting small identical squares from each corner of a 12 -by-12-in. sheet of tin and bending up the sides. If each corner square is x inches on a side, the volume of the box (in in. ³) is given by $V ( x ) = 144 x - 48 x ^ { 2 } + 4 x ^ { 3 }$. By sketching the graph of V(x) , estimate what values of x result in a box with a volume greater than 64 in³ . A) $1.0 \text { in. } \leq x \leq 3.0 \text { in }$ B) $0.1 \text { in. } \leq x \leq 5.0 \text { in }$ C) $0.54 \text { in. } \leq x \leq 4 \text { in. }$ D) $1.5 \text { in. } \leq x \leq 2.5 \text { in. }$

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Quizplus · Accepted Answer

The volume function $V(x) = 144x - 48x^2 + 4x^3$ represents a cubic equation, which typically has two turning points. For a box volume to be greater than 64 cubic inches, we look for the range of $x$ values where $V(x) > 64$. Without the exact graph, we rely on the nature of cubic functions and the given options. Option C suggests a reasonable range for $x$ that would allow for a significant volume (greater than 64 cubic inches) before the volume starts to decrease again due to the sides of the box becoming too tall and narrow. This range captures the behavior of a cubic function's increase to a maximum and then decrease, fitting the physical constraints of the problem.

An Open-Top Box Is to Be Made by Cutting Small V(x)=144x−48x2+4x3V ( x ) = 144 x - 48 x ^ { 2 } + 4 x ^ { 3 }V(x)=144x−48x2+4x3

An Open-Top Box Is to Be Made by Cutting Small $V ( x ) = 144 x - 48 x ^ { 2 } + 4 x ^ { 3 }$