1. **State the problem:** We have a cylindrical tin can with height function $h(t) = 3t^2 + 0.5$ for $4 \leq t \leq 14$. We want to find the height when $t$ is given. 2. **Calculate the height for a specific $t$:** For example, if $t=4$, substitute into the function: $$h(4) = 3(4)^2 + 0.5 = 3 \times 16 + 0.5 = 48 + 0.5 = 48.5$$ 3. **Interpret the inverse function $h^{-1}$:** The inverse function $h^{-1}$ reverses the roles of input and output, so given a height $y$, it returns the time $t$ such that $h(t) = y$. 4. **Range of $h$:** Since $h(t) = 3t^2 + 0.5$ and $t$ ranges from 4 to 14, the minimum height is at $t=4$: $$h(4) = 48.5$$ The maximum height is at $t=14$: $$h(14) = 3(14)^2 + 0.5 = 3 \times 196 + 0.5 = 588 + 0.5 = 588.5$$ So the range $Y$ is: $$[48.5, 588.5]$$ 5. **Quadratic function $y = ax^2 + bx + c$ with vertex $(0.5, 12.5)$ and roots at $(0,0)$ and $(2,0)$:** 6. **Find $p$:** The problem does not define $p$ explicitly, but assuming $p$ relates to the vertex or coefficients, we proceed to find $a$, $b$, and $c$. 7. **Use roots to find $c$ and relation between $a$ and $b$:** Since roots are at $x=0$ and $x=2$, the quadratic can be written as: $$y = a x (x - 2) = a(x^2 - 2x)$$ So $c=0$. 8. **Use vertex to find $a$ and $b$:** Vertex formula for $x$-coordinate is: $$x_v = -\frac{b}{2a} = 0.5$$ Since $y = a x^2 + b x + 0$, then $b = -2a x_v = -2a(0.5) = -a$ 9. **Use vertex $y$-coordinate:** $$y_v = a x_v^2 + b x_v = a (0.5)^2 + b (0.5) = a (0.25) + b (0.5) = 12.5$$ Substitute $b = -a$: $$0.25 a + 0.5 (-a) = 12.5 \Rightarrow 0.25 a - 0.5 a = 12.5 \Rightarrow -0.25 a = 12.5 \Rightarrow a = -50$$ 10. **Find $b$:** $$b = -a = -(-50) = 50$$ 11. **Equation of axis of symmetry:** $$x = x_v = 0.5$$ **Final answers:** - Height at $t=4$ is $48.5$ cm. - Range of $h$ is $[48.5, 588.5]$. - $p$ is not explicitly defined. - $a = -50$, $b = 50$, $c = 0$. - Axis of symmetry is $x = 0.5$.