Question

Read and explain 2 parts A and B and complete Your Turn #5 below (adapted from Lesson 6.2). Find the polynomial function modeling the desired relationship. Brent runs a small toy store specializing in wooden toys. From 2000 through 2012, the number of toys Brent made can be modeled by (N(x) = 0.7x^2 – 2x + 23), and the average cost to make each toy can be modeled by (C(x) = -0.004x^2 – 0.08x + 25), where (x) is the number of years since 2000. Write a polynomial that can be used to model Brent's total cost for making the toys, (T(x)), for those years.

a) (T(x) = 0.7x^2 - 2x + 23 - (0.004x^2 + 0.08x - 25))
b) (T(x) = 0.696x^2 - 2.08x + 48)
c) (T(x) = 0.696x^2 - 2.08x + 48 - (0.004x^2 + 0.08x - 25))
d) (T(x) = -0.004x^2 - 0.36x + 48)

Answer 1

To calculate Brent's total cost for making the toys (T(x)), you need to multiply the number of toys made (N(x)) by the average cost per toy (C(x)). None of the provided options (a-d) correctly show this multiplication, as the operations indicated are not multiplication. The correct polynomial for T(x) would be found by distributing each term in N(x) by each term in C(x), combining like terms, then writing down the resulting polynomial.

Step-by-step explanation:

The question is asking to find a polynomial function that can model the total cost of making toys (T(x)) from the year 2000 through 2012 for Brent's toy store, by combining the polynomial functions for the number of toys made (N(x)) and the average cost to make each toy (C(x)). To find the total cost, T(x), we have to multiply the number of toys made (N(x)) by the average cost per toy (C(x)).

The correct polynomial is found by multiplying N(x) = 0.7x² – 2x + 23 by C(x) = -0.004x² – 0.08x + 25. This multiplication leads us to calculate T(x) as:

T(x) = (0.7x² – 2x + 23) * (-0.004x² – 0.08x + 25)

However, none of the provided options (a-d) correctly represent the multiplication of these two polynomials. Options (a), (c) and (d) are incorrect because they show subtraction instead of multiplication. Option (b) seems to be a modification of N(x) rather than a proper multiplication of N(x) and C(x).

To arrive at the correct answer, you should follow these steps:

1. Distribute each term in N(x) by each term in C(x).
2. Combine like terms.
3. Write down the resulting polynomial for T(x).