A popular model of carry-on luggage has a length that is 10 inches greater than its

Chapter 5, Problem 119

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A popular model of carry-on luggage has a length that is 10 inches greater than its depth. Airline regulations require that the sum of the length, width, and depth cannot exceed 40 inches. These conditions, with the assumption that this sum is 40 inches, can be modeled by a function that gives the volume of the luggage, V, in cubic inches, in terms of its depth, x, in inches. V(x) = x (x+10) [40-(x+x+10)] V(x) = x(x+10)(30-2x) V = olume depth length width: 40 (depth + length) a. Perform the multiplications in the formula for V(x) and express the formula in standard form. b. Use the functions formula from part (a) and the Leading Coefficient Test to determine the end behavior of its graph. c. Does the end behavior to the right make this function useful in modeling the volume of carry-on luggage as its depth continues to increase? d. Use the formula from part (a) to find V(10). Describe what this means in practical terms. e. The graph of the function modeling the volume of carry-on luggage is shown below. Identify your answer from part (d) as a point on the graph. x y 5 1 10 5 20 25 1000 2000 3000 1000 2000 25 20 15 10 5 y = V(x) f. Use the graph to describe a realistic domain, x, for the volume function, where x represents the depth of the carry-on luggage. Use interval notation to express this realistic domain.