A business analyst models a company's weekly net profit, N(x), in dollars, as the differen… — ACT

A business analyst models a company's weekly net profit, $N(x)$ , in dollars, as the difference between its weekly revenue, $R(x) = (2x + 5)(30 - x)$ , and its weekly production cost, $C(x) = (x - 4)^2 + 150$ , where $x$ represents the number of items sold. When $N(x)$ is simplified and written in standard form as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of the coefficient $b$ ?

Answer: 63

Answer

The coefficient

b

63

Expanding the revenue polynomial gives

R(x) = -2x^2 + 55x + 150

and the cost polynomial gives

C(x) = x^2 - 8x + 166

. Subtracting them gives

N(x) = (-2x^2 + 55x + 150) - (x^2 - 8x + 166) = -3x^2 + 63x - 16

. The coefficient of the

x

term, which corresponds to

b

, is

63

Step-by-Step Solution

Expand the revenue expression

R(x) = (2x + 5)(30 - x)

using polynomial multiplication.

R(x) = -2x^2 + 55x + 150

To find the polynomial representing revenue in standard form.

Expand the cost expression

C(x) = (x - 4)^2 + 150

using binomial squaring.

C(x) = x^2 - 8x + 166

To find the polynomial representing cost in standard form.

Subtract the cost polynomial from the revenue polynomial:

N(x) = R(x) - C(x)

, distributing the negative sign through all terms of the cost polynomial.

N(x) = -3x^2 + 63x - 16

To determine the net profit polynomial

N(x)

in standard form.

Identify the coefficient

b

of the

x

term in the standard form

ax^2 + bx + c

b = 63

To answer the specific question asking for the coefficient of the middle term.

Key Concept

Polynomial subtraction and expansion of algebraic expressions.

Question

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