From 1980 to 2002, the number T(in millions) of households in the United States with televisions and the percent P of those households with VCRs can be modeled by

T=1.22x+76.9 and P=-0.205x^2+8.36x+1.98

where x is the total number of years since 1980. Write a polynomial model for the total number of U.S. households with both televisions and VCRs.

To model the total number of U.S. households with both televisions and VCRs, we need to combine the given models for \( T \) and \( P \).

1. **Given Models**:
   - \( T = 1.22x + 76.9 \)
   - \( P = -0.205x^2 + 8.36x + 1.98 \)

   Here, \( T \) represents the total number of households with televisions (in millions), and \( P \) represents the percentage of those households that have VCRs. \( x \) is the number of years since 1980.

2. **Calculation**:
   To find the total number of households with both televisions and VCRs, we need to calculate \( \frac{P}{100} \times T \), which is the product of the percentage (in decimal form) and the total number of households with televisions.

   \[
   \text{Households with both} = \left( \frac{P}{100} \right) \times T
   \]

   Substitute the given models into this expression:

   \[
   \text{Households with both} = \left( \frac{-0.205x^2 + 8.36x + 1.98}{100} \right) \times (1.22x + 76.9)
   \]

3. **Simplify**:
   \[
   \text{Households with both} = \left( -0.00205x^2 + 0.0836x + 0.0198 \right) \times (1.22x + 76.9)
   \]

   Now, expand and simplify the product:

   \[
   \begin{align*}
   \text{Households with both} &= (-0.00205x^2 + 0.0836x + 0.0198)(1.22x + 76.9) \\
   &= -0.00205x^2 \cdot 1.22x + -0.00205x^2 \cdot 76.9 + 0.0836x \cdot 1.22x + 0.0836x \cdot 76.9 + 0.0198 \cdot 1.22x + 0.0198 \cdot 76.9
   \end{align*}
   \]

   \[
   \begin{align*}
   &= -0.00205 \cdot 1.22x^3 + -0.00205 \cdot 76.9x^2 + 0.0836 \cdot 1.22x^2 + 0.0836 \cdot 76.9x + 0.0198 \cdot 1.22x + 0.0198 \cdot 76.9
   \end{align*}
   \]

4. **Combine like terms**:
   \[
   \begin{align*}
   &= -0.002501x^3 + (-0.00205 \cdot 76.9 + 0.0836 \cdot 1.22)x^2 + (0.0836 \cdot 76.9 + 0.0198 \cdot 1.22)x + 0.0198 \cdot 76.9 \\
   &= -0.002501x^3 + (-0.157645 + 0.102792)x^2 + (6.43044 + 0.024156)x + 1.52362 \\
   &= -0.002501x^3 - 0.054853x^2 + 6.454596x + 1.52362
   \end{align*}
   \]

5. **Final Polynomial Model**:
   \[
   \boxed{N = -0.002501x^3 - 0.054853x^2 + 6.454596x + 1.52362}
   \]

This polynomial \( N \) models the total number of U.S. households with both televisions and VCRs (in millions) from 1980 to 2002.

Would you like further details or have any questions?

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**Tip:** Always check your expanded polynomial terms for combining like terms correctly to avoid any mistakes in coefficients.

Learn how to model the total number of U.S. households with both televisions and VCRs from 1980 to 2002 using polynomial functions. This problem involves polynomial equations and their applications in real-world data modeling. Suitable for high school students.

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