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Part A. Describe a relationship modeled by the function f(x) = 4x³ - 72x² + 320x, and explain how the function models the relationship.
Part B. Identify and interpret the key features of the function in the context of the situation described in part A.

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Answer:A relationship modeled by the function f(x) = 4x³ - 72x² + 320x is the volume of a right prism whose dimensions are 4 times a desired length, 10 units less that such desired length, and 8 units less than the same desired length.

Explanation:

To find a relationship modeled by the given function it is recommendable to factor it.

The function is: f(x)=4x3- 72 x2+320x

The first step to factor it is to extract common factor 4x: f (x)=4x(x2-18=80)

The second step is to factor the quadratic trinomial.

That is made by writing it as a product of two binomials, for which the two constant terms add up - 18 and their product is 80. Those terms are -10 and - 8; so the two factors are (x - 10) and (x - 8), and the factored form is: f(x)=(4x)(x-10)(x-8)

Then, a relationship modeled by that polynomial is the volume of right prism whose dimensions are 4 times a desired length, 10 units less that such desired length, and 8 units less than the same desired length.

x is the desired (unknown) length

4x is 4 times the desired length

x - 10 is 10 less than the desired length

x - 8 is 8 less than the desired length

Thus, the volume of the prism is the product of the three factors:

Volume = (4x)(x-10)(x-8)

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