Answer:
A) The vertex form of a quadratic function is given by:
y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola and a is the coefficient that determines the shape of the parabola.
The archway has a width of 280 cm, so the distance from the origin to the highest point is 140 cm. The vertex of the parabola is at the highest point of the archway, which is (140, 216).
To find the coefficient a, we can use the fact that the archway also passes through the points (0,0) and (280,0). Substituting these values into the vertex form of the equation, we get:
0 = a(0 - 140)^2 + 216
0 = 19600a + 216
a = -216/19600 = -0.011
Therefore, the equation of the archway in vertex form is:
y = -0.011(x - 140)^2 + 216
B) We want to find the height of the archway at a point that is 50 cm from its outer edge. We can set x = 50 in the equation we found in part A:
y = -0.011(50 - 140)^2 + 216
y = -0.011(-90)^2 + 216
y = -0.011(8100) + 216
y = -89.1
Rounding to the nearest tenth, the height of the archway at a point that is 50 cm from its outer edge is approximately 89.1 cm.