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To solve number 28, we need to write a polynomial model for the total number of U.S. households with both televisions and VCRs given the following models:

\[
T = 1.22x + 76.9
\]
\[
P = -0.205x^2 + 8.36x + 1.98
\]

Where:
- \( T \) is the number of households with televisions (in millions).
- \( P \) is the percentage of those households with VCRs.
- \( x \) is the number of years since 1980.

To find the total number of households with both televisions and VCRs, we need to calculate \( \frac{P}{100} \times T \).

1. Convert \( P \) to a decimal by dividing by 100:

\[
\frac{P}{100} = \frac{-0.205x^2 + 8.36x + 1.98}{100} = -0.00205x^2 + 0.0836x + 0.0198
\]

2. Multiply this by \( T \):

\[
N = \left(-0.00205x^2 + 0.0836x + 0.0198\right) \times \left(1.22x + 76.9\right)
\]

3. Distribute and simplify:

\[
N = \left(-0.00205x^2 + 0.0836x + 0.0198\right) \times \left(1.22x + 76.9\right)
\]

Expanding the product:

\[
N = (-0.00205x^2)(1.22x) + (-0.00205x^2)(76.9) + (0.0836x)(1.22x) + (0.0836x)(76.9) + (0.0198)(1.22x) + (0.0198)(76.9)
\]

Calculating each term:

\[
N = -0.00205 \times 1.22 x^3 + -0.00205 \times 76.9 x^2 + 0.0836 \times 1.22 x^2 + 0.0836 \times 76.9 x + 0.0198 \times 1.22 x + 0.0198 \times 76.9
\]

\[
N = -0.002501 x^3 - 0.157645 x^2 + 0.102792 x^2 + 6.43044 x + 0.024156 x + 1.52362
\]

Combine like terms:

\[
N = -0.002501 x^3 - 0.054853 x^2 + 6.454596 x + 1.52362
\]

Therefore, the polynomial model for the total number of U.S. households with both televisions and VCRs is:

\[
\boxed{N = -0.002501 x^3 - 0.054853 x^2 + 6.454596 x + 1.52362}
\]

Would you like further details or have any questions?

Here are 5 questions you might be interested in:
1. How do I find the vertex of a quadratic function?
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**Tip:** When dealing with polynomial expansions, it’s useful to write each term carefully and check your work to avoid errors in coefficients.

Explore the polynomial model for estimating the number of U.S. households with both televisions and VCRs over time. Learn how to apply polynomial regression and analyze data trends effectively.

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