1. **State the problem:** We need to find a quadratic function modeling an archway with maximum height 4 m and maximum width 4 m, then check if an object 3 m wide and 1.6 m tall fits through it. 2. **Model the archway:** The archway is a parabola opening downward with zeros at $x=0$ and $x=4$, and vertex at $x=2$ with height 4. 3. **Write the quadratic function:** Since zeros are at 0 and 4, the function can be written as $$y = a x (4 - x)$$ where $a$ is a constant. 4. **Find $a$ using the vertex:** The vertex is at $x=2$, and $y=4$ there. Calculate: $$y(2) = a \cdot 2 \cdot (4 - 2) = 4a = 4$$ So, $$a = 1$$ 5. **Final function:** $$y = x(4 - x) = 4x - x^2$$ 6. **Check if the object fits:** The object is 3 m wide and 1.6 m tall. The object width 3 m means it spans from $x=0.5$ to $x=3.5$ (centered under the arch). Calculate the arch height at $x=0.5$ and $x=3.5$: $$y(0.5) = 4(0.5) - (0.5)^2 = 2 - 0.25 = 1.75$$ $$y(3.5) = 4(3.5) - (3.5)^2 = 14 - 12.25 = 1.75$$ Since the arch height at these points is 1.75 m, which is greater than the object height 1.6 m, the object fits. **Answer:** - a) The function modeling the archway is $$y = 4x - x^2$$. - b) The object 3 m wide and 1.6 m tall will fit through the archway.