1. **State the problem:** We want to find an equation that models the height $H$ (in cm) of a child as a function of age $A$ (in years) using the given data points: $(2, 84.5)$, $(3, 90.5)$, $(5, 105.0)$, $(7, 119.0)$, $(8, 125.5)$, and $(12, 151.5)$. 2. **Choose a model:** Since height generally increases approximately linearly with age in childhood, we try a linear model: $$H = mA + b$$ where $m$ is the rate of growth (cm per year) and $b$ is the height at age 0. 3. **Calculate slope $m$:** Using two points, for example $(2, 84.5)$ and $(12, 151.5)$, $$m = \frac{151.5 - 84.5}{12 - 2} = \frac{67}{10} = 6.7$$ cm/year. 4. **Calculate intercept $b$:** Using point $(2, 84.5)$, $$84.5 = 6.7 \times 2 + b \implies b = 84.5 - 13.4 = 71.1$$ 5. **Write the equation:** $$H = 6.7A + 71.1$$ 6. **Check with another point:** For $A=5$, $$H = 6.7 \times 5 + 71.1 = 33.5 + 71.1 = 104.6$$ which is close to the actual 105.0 cm, confirming the model fits well. 7. **Limitations:** This linear model is valid only for the age range given (approximately 2 to 12 years). It may not accurately predict height outside this range because growth rates change with age. **Final answer:** $$\boxed{H = 6.7A + 71.1}$$ where $H$ is height in cm and $A$ is age in years, valid for $2 \leq A \leq 12$.