Question

Describe a relationship modeled by the function f(x) = 4x3 − 72x2 + 320x

Answer 1

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:

• $f(x)=4x^3-72x^2+320x$

The first step to factor it is to extract common factor 4x:

• $f(x)=4x(x^2-18x+80)$

The second step is to factor the quadratic trinomial.

That is made by writting 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)