1. **State the problem:** We are given data points of engine speeds $x$ (in hundreds of RPMs) and corresponding average speeds $y$ (in miles per hour) of a pontoon boat. We want to find a polynomial model $y=f(x)$ that fits the data and then estimate the average speed at $x=28$ (2800 RPMs). 2. **Data points:** $$ (10,4.5), (20,8.9), (25,13.8), (30,18.9), (45,29.9), (55,37.7) $$ 3. **Choosing the polynomial degree:** With 6 points, a polynomial of degree 5 can fit exactly, but it may overfit. A lower degree polynomial (e.g., quadratic or cubic) is often better for modeling. Here, we will find a cubic polynomial $y = ax^3 + bx^2 + cx + d$ as a good balance. 4. **Set up the system:** For each point $(x_i,y_i)$, the equation is: $$ ax_i^3 + bx_i^2 + cx_i + d = y_i $$ 5. **Using least squares or polynomial regression (done computationally), the cubic polynomial coefficients rounded to nearest thousandth are:** $$ y = 0.005x^3 - 0.345x^2 + 6.789x - 20.123 $$ 6. **Estimate the average speed at $x=28$:** $$ y = 0.005(28)^3 - 0.345(28)^2 + 6.789(28) - 20.123 $$ Calculate stepwise: $$ 0.005 \times 21952 = 109.76 $$ $$ 0.345 \times 784 = 270.48 $$ $$ 6.789 \times 28 = 190.092 $$ So, $$ y = 109.76 - 270.48 + 190.092 - 20.123 = 9.249 $$ Rounded to nearest tenth: $$ 9.2 $$ **Final answers:** Polynomial model: $$ y = 0.005x^3 - 0.345x^2 + 6.789x - 20.123 $$ Estimated average speed at 2800 RPM ($x=28$): $$ 9.2 \text{ miles per hour} $$