Question

brockton school pre-calculus 11 page 7 of 7 4. a stone bridge has the shape of a parabolic arch, as shown. determine a quadratic function to represent the shape of the arch if the origin: a. is at the top of the opening under the bridge b. is on the ground at the midpoint of the opening c. is at the base of the bridge on the right side of the opening d. is on the left side at the top surface of the bridge hint: the positions of the origin are indicated on the diagram below. coordinates of the origin are (0,0) diagram shows a black rectangular shape with a white semicircular (parabolic) cutout. dimensions: 56 ft (vertical total), 40 ft (vertical dashed), 12 ft (vertical from origin to curve), 15 ft (horizontal from curve to right side), with origin positions marked as a, b, c, d.

Explanation:

Step1: Identify the vertex form of a parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the diagram, the vertex of the parabola (the top of the opening) is at \((0, 12)\) if we consider the origin at the midpoint of the opening's base? Wait, no, let's re-examine. Wait, the hint says coordinates of the origin are \((0,0)\). Wait, the options: let's check the options. The options are about where the origin is. Let's analyze the diagram. The diagram has a parabolic arch (the white part) with a black rectangle (the bridge structure). The vertical distance from the base (maybe the bottom of the black part) to the top of the arch? Wait, the labels: 40 ft, 56 ft, 12 ft, 15 ft. Wait, the origin's coordinates are \((0,0)\) as per the hint. Let's check the options:

a. is at the top of the opening under the bridge: If origin is at the top of the opening (the vertex of the parabola), then the parabola opens downward. Let's see the vertical measurements. The distance from the origin (top of opening) to the base (bottom of the black part) would be? Wait, 12 ft is the distance from the origin (maybe) to some point? Wait, the diagram has a dashed line with 12 ft, 40 ft, 56 ft. Wait, maybe the origin is at the midpoint of the opening's base (option c)? No, option c is "is at the base of the opening at the midpoint of the opening". Wait, let's think about the parabola's equation. If the origin is at the midpoint of the opening's base (the bottom of the parabolic arch), then the parabola opens upward? But the arch is a downward opening parabola (since it's a bridge arch, the opening is a downward parabola). Wait, maybe the origin is at the top of the opening (option a). Let's check the vertical distances. The total height of the bridge (black part) is 56 ft, and the height from the base (bottom of black) to the top of the arch (white) is 40 ft? Wait, no, 56 - 40 = 16? No, maybe the 12 ft is the distance from the origin (top of opening) to the top of the bridge? Wait, the black part's height is 56 ft, and the white arch has a height of 40 ft? No, maybe the origin is at the midpoint of the opening's base (option c). Wait, the opening is the white parabolic arch. The base of the opening would be the bottom of the arch (the midpoint of the base of the arch). If the origin is at the base of the opening (the bottom) at the midpoint, then the parabola opens upward. But the arch is a bridge arch, so it should open downward. Wait, maybe the origin is at the top of the opening (option a), so the parabola opens downward, with vertex at \((0,0)\)? No, the 12 ft is a vertical measurement. Wait, the key is to determine which option is correct based on the diagram's coordinates. Let's re-read the hint: "Coordinates of the origin are (0,0)". The diagram has labels: 15 ft (horizontal), 12 ft (vertical from origin to a point on the arch), 40 ft (vertical from origin to another point), 56 ft (total height of bridge). Let's assume the origin is at the top of the opening (the vertex of the parabola), so the parabola opens downward. Then the equation would be \( y = -ax^2 \). The width of the opening (the horizontal span of the parabola) is 15 ft on each side? Wait, the horizontal distance from the origin (center) to the right side of the opening is 15 ft (since the black part is 15 ft wide, so the opening's width is 30 ft? No, the black part is 15 ft, so the opening is 15 ft on each side? Wait, maybe the origin is at the midpoint of the opening's base (option c). Wait, no, let's check the options again:

a. is at the top of the opening under the bridge: If origin is at the top of the opening (the vertex of the parabola), then the parabola opens downward. The vertical distance from the origin (top) to the base of the opening (bottom of the arch) would be, say, 40 ft (from the diagram's 40 ft label). Then the equation would be \( y = -ax^2 + k \), but if origin is at (0,0), then k=0, so \( y = -ax^2 \). The point on the arch is (15, -12)? Wait, no, the 12 ft is the distance from the origin to a point on the arch. Wait, maybe the origin is at the top of the opening (option a), so the parabola has vertex at (0,0) (origin), opens downward, and passes through (15, -12) (since 12 ft below the origin). Then the equation would be \( y = -ax^2 \). Plugging in (15, -12): -12 = -a(15)^2 → 12 = 225a → a = 12/225 = 4/75. But let's check the other options. Option c: "is at the base of the opening at the midpoint of the opening". If origin is at the base (bottom of the arch) at the midpoint, then the parabola opens upward, with vertex at (0,0), and passes through (15, 12) (12 ft above the origin). Then equation is \( y = ax^2 \), plugging (15,12): 12 = a(225) → a = 12/225 = 4/75. But the bridge's total height is 56 ft, so from the base (origin) to the top of the bridge (black part) is 56 ft, and the height from the base to the top of the arch is 40 ft? Wait, 56 - 40 = 16? No, maybe the 56 ft is the total height of the bridge (black part), and the arch (white) has a height of 40 ft, so the distance from the top of the arch to the top of the bridge is 56 - 40 = 16 ft? No, this is confusing. Wait, the key is to determine the origin's position. The options are about where the origin is:

a. top of the opening (under the bridge, i.e., the vertex of the parabola)

b. on the ground at the midpoint of the opening: but the diagram doesn't show ground, it shows the bridge structure.

c. base of the opening at the midpoint: base of the opening (bottom of the arch) at midpoint.

d. on the left side at the top surface: left side, top surface.

Looking at the diagram, the origin is at (0,0), and there's a point 12 ft below the origin (since the arrow is downward from origin to a point on the arch, labeled 12 ft). The horizontal distance from origin to the right side of the opening is 15 ft (labeled 15 ft). So the origin is at the top of the opening (the vertex of the parabola), because the parabola opens downward (since the arch is a bridge arch, the opening is a downward parabola), so the vertex is at the top (origin), and the parabola opens downward. So option a is correct.

Step2: Confirm with the parabola's direction

A bridge arch is a downward - opening parabola, so its vertex (the highest point of the opening) is the top of the opening. If the origin is at \((0,0)\) and is at the top of the opening, this matches the vertex of a downward - opening parabola. The other options: option b (ground) is not indicated in the diagram (the diagram shows the bridge structure, not ground), option c (base of opening) would be the bottom of the parabola (vertex of upward - opening), which doesn't match a bridge arch, option d (left side top surface) would not be the vertex of the parabolic opening. So the origin is at the top of the opening under the bridge.

Answer:

a. is at the top of the opening under the bridge