Answer:
Explanation:
The equation h = -0.0159d² + 290 models the curve of the natural stone bridge, where h is the height of the curve in feet and d is the width of the curve in feet.
To find the vertex of the curve, we need to use the formula x = -b / 2a, where x represents the x-coordinate of the vertex and a and b are coefficients of the quadratic equation. In this case, the quadratic equation is h = -0.0159d² + 290, so a = -0.0159 and b = 0. To find the vertex, we plug these values into the formula:
x = -b / 2a = 0 / (2 * (-0.0159)) = 0
So the x-coordinate of the vertex is 0, which means the vertex is at the point (0, 290).
The vertex tells us that the highest point of the bridge (the vertex) is at a width of 0 feet. This means that the bridge is tallest at its center, and the height decreases as the width increases in either direction. Additionally, since the coefficient of the squared term is negative, the parabolic curve is downward-facing, which indicates that the bridge curves downward as it extends outward.
To find the span of the rainbow bridge, we need to find the width of the bridge where the height is 0. This represents the point where the bridge intersects with the river. To do this, we set h = 0 and solve for d:
0 = -0.0159d² + 290
0.0159d² = 290
d² = 290 / 0.0159
d ≈ 77.36
So the width of the bridge where the height is 0 is approximately 77.36 feet. Therefore, the span of the rainbow bridge is approximately 154.72 feet (twice the width at the point where the height is 0).