# A box in the shape of a rectangular prism is shown.  l=12-w, height is 5.  Determine the maximum volume of the box and the reasonable domain and range of this function. 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....

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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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