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The volume of the rectangular prism is determined by the product of its length, width and height. Is the width is w, the volume will be the function
V(w) = w*l*H = w(12 - w)*5 = 5w(12-w).
`V(w) = 5w(12-w) = 60w - 5w^2` .
This is a quadratic function. The general form of this function is `aw^2 + bw + c` and its graph is a parabola.
In case of V(w), it is an upside down parabola because the coefficient in front of `w^2` a = -5. This means the maximum value of this function (i.e. maximum volume) is attained at the vertex.
The x-coordinate of the vertex of a general parabola is given by equation `-b/(2a)` .
For V(w), `-b/(2a) = -60/(2*(-5)) = 6` .
So the volume will be maximal when the width is 6. Then, length is l = 12 - 6 = 6 and the volume is V= 6*6*5 = 180.
The maximum volume of the box is 180.
Generally, the domain of the quadratic function is all real numbers, but since the variable w represents the width of a box, it has to be positive. So, the domain is w>0.
The range is a possible volume. Since the maximum volume is 180, and volume cannot be negative, the range is `0<V(w) <=180` .
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