1. **State the problem:** We are given a table of heights $y$ (in feet) at different times $x$ (in seconds) and asked to predict the height after 5 seconds. 2. **Analyze the data:** The table is: $$\begin{array}{c|c} \text{Seconds } x & \text{Height } y \\ \hline 0 & 3 \\ 0.5 & 39 \\ 1 & 67 \\ 1.5 & 87 \\ 2 & 99 \end{array}$$ 3. **Find the pattern:** Let's check if the height follows a quadratic pattern since the increase is not constant. 4. **Use quadratic model:** Assume $y = ax^2 + bx + c$. 5. **Use points to form equations:** At $x=0$, $y=3$ gives: $$3 = a\cdot0^2 + b\cdot0 + c \Rightarrow c = 3$$ At $x=1$, $y=67$ gives: $$67 = a\cdot1^2 + b\cdot1 + 3 \Rightarrow a + b + 3 = 67 \Rightarrow a + b = 64$$ At $x=2$, $y=99$ gives: $$99 = a\cdot4 + b\cdot2 + 3 \Rightarrow 4a + 2b + 3 = 99 \Rightarrow 4a + 2b = 96$$ 6. **Solve the system:** From $a + b = 64$, multiply by 2: $$2a + 2b = 128$$ Subtract from $4a + 2b = 96$: $$4a + 2b - (2a + 2b) = 96 - 128 \Rightarrow 2a = -32 \Rightarrow a = -16$$ Then from $a + b = 64$: $$-16 + b = 64 \Rightarrow b = 80$$ 7. **Write the quadratic equation:** $$y = -16x^2 + 80x + 3$$ 8. **Predict height at $x=5$ seconds:** $$y = -16(5)^2 + 80(5) + 3 = -16 \times 25 + 400 + 3 = -400 + 400 + 3 = 3$$ 9. **Interpretation:** The height after 5 seconds is predicted to be 3 feet. This suggests the object rises and then falls back to the initial height at 5 seconds.