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The volume of a rectangular box is given by the expression V = (120 − 6w)w2, where w is measured in inches.
a.
What is a reasonable domain for the function in this situation? Express the domain as an inequality, in interval notation, and in set notation.
b.
Sketch a graph of the function over the domain that you found. Include the scale on each axis.
c.
Use a graphing calculator to find the coordinates of the maximum point of the function.
What is the width of the box, in inches, that produces the maximum volume?
a. To determine the reasonable domain for the given function, we need to consider the real-world context of the problem. In this case, the function represents the volume of a rectangular box, and the width of the box is represented by the variable w.
Since width cannot be negative, we can conclude that the domain for this function must be all non-negative values of w. In other words, w ≥ 0.
Expressing the domain in interval notation, we can write it as [0, ∞).
In set notation, the domain can be written as the set of real numbers w such that w ≥ 0: {w ∈ ℝ | w ≥ 0}.
b. To sketch a graph of the function, we can plot points on a coordinate plane using different values of w and calculate the corresponding values of V.
Let's choose several values of w and calculate the corresponding values of V:
When w = 0, V = 0.
When w = 1, V = 114.
When w = 2, V = 224.
When w = 3, V = 330.
And so on...
Plotting these points on a graph and connecting them, we can obtain a graph of the function. Remember to label the axes with appropriate scales.
c. To find the coordinates of the maximum point of the function, we can use a graphing calculator. Enter the function V = (120 − 6w)w^2 into the calculator and graph it over the domain [0, ∞). The graphing calculator will display the graph of the function.
Look for the highest point on the graph, which represents the maximum volume. Note its x-coordinate or the corresponding width value, which will give the width of the box that produces the maximum volume.
You can use the calculator's trace function or find the coordinates explicitly if the calculator provides that information.
Once you have the x-coordinate or width value, you can substitute it back into the original expression for V to find the maximum volume.