1. **State the problem:** We have a parabolic arch represented by the polynomial $$p(x) = -0.0025x^2 - 0.025x + 136$$. (i) Write the coordinates of point A (the highest point on the parabola). (ii) Find the span of the arch (distance between points Q and P where the parabola intersects the x-axis). (iii) (b) Find the values of $$p(x)$$ at $$x=100$$ and $$x=-100$$ and check if they are the same. --- 2. **Find point A (vertex of the parabola):** The vertex of a parabola $$y = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. Here, $$a = -0.0025$$ and $$b = -0.025$$. Calculate: $$x_A = -\frac{-0.025}{2 \times -0.0025} = -\frac{-0.025}{-0.005} = -5$$ Now find $$p(-5)$$: $$p(-5) = -0.0025(-5)^2 - 0.025(-5) + 136 = -0.0025(25) + 0.125 + 136 = -0.0625 + 0.125 + 136 = 136.0625$$ So, point A is $$(-5, 136.0625)$$. --- 3. **Find the span of the arch:** The span is the distance between points Q and P on the x-axis. Given: $$Q = (-238.5, 0)$$ $$P = (228.5, 0)$$ Span = $$228.5 - (-238.5) = 228.5 + 238.5 = 467$$ --- 4. **Find $$p(100)$$ and $$p(-100)$$:** Calculate $$p(100)$$: $$p(100) = -0.0025(100)^2 - 0.025(100) + 136 = -0.0025(10000) - 2.5 + 136 = -25 - 2.5 + 136 = 108.5$$ Calculate $$p(-100)$$: $$p(-100) = -0.0025(-100)^2 - 0.025(-100) + 136 = -0.0025(10000) + 2.5 + 136 = -25 + 2.5 + 136 = 113.5$$ Since $$p(100) = 108.5$$ and $$p(-100) = 113.5$$, they are not the same. --- **Final answers:** (i) Point A coordinates: $$(-5, 136.0625)$$ (ii) Span of the arch: $$467$$ units (iii) (b) $$p(100) = 108.5$$ and $$p(-100) = 113.5$$, so values are not the same.