Question

the main cable of a suspension bridge forms a parabola modeled by the equation $y = a(x - h)^2 + k$ where $y$ is the height in feet of the cable above the road, $x$ is the horizontal distance in feet from the right bridge support, $a$ is a constant, and $(h, k)$ is the parabola’s vertex. what is the maximum and minimum height of the bridge modeled by the equation $y = 0.005(x - 60)^2 + 8$?

• maximum height = 100 feet and minimum height = 26 feet
• maximum height = 100 feet and minimum height = 8 feet
• maximum height = 60 feet and minimum height = 26 feet
• maximum height = 26 feet and minimum height = 8 feet

Explanation:

Step1: Analyze the parabola equation form

The given equation is \( y = 0.005(x - 60)^2 + 8 \), which is in the vertex form of a parabola \( y=a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \(a = 0.005\), \(h = 60\), and \(k = 8\). Since \(a=0.005>0\), the parabola opens upwards, so the vertex \((60,8)\) is the minimum point (the lowest point of the parabola).

Step2: Determine the minimum height

The minimum height occurs at the vertex, so the minimum height \(y = 8\) feet (when \(x = 60\), since the vertex is at \((60,8)\)).

Step3: Find the maximum height (considering the context of a bridge)

For a suspension bridge, the main cable is attached to the bridge supports. The right bridge support is at \(x = 0\) (since \(x\) is the horizontal distance from the right bridge support). Let's substitute \(x = 0\) into the equation to find the height at the right support (which will be a maximum height in the context of the bridge, as the parabola opens upwards and the supports are at the ends).

Substitute \(x = 0\) into \(y=0.005(x - 60)^2+8\):

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Wait, but maybe the left support is at \(x = 120\) (since the vertex is at \(x = 60\), the distance between the two supports is \(120\) feet, so \(x = 0\) and \(x = 120\) are the two supports). Let's check \(x = 120\):

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Wait, but the options have a maximum height of 26? Wait, maybe I made a mistake. Wait, the options have "maximum height = 26 feet and minimum height = 8 feet" as one of the options. Let's re - evaluate.

Wait, the parabola \(y = 0.005(x - 60)^2+8\) opens upwards. The minimum height is at the vertex (\(y = 8\)). The maximum height in the context of the bridge (the height at the supports) is when \(x = 0\) or \(x = 120\) (assuming the span of the bridge is from \(x = 0\) to \(x = 120\), with the vertex at \(x = 60\)). But when we calculate \(y\) at \(x = 0\) or \(x = 120\), we get \(y=0.005\times(0 - 60)^2+8=0.005\times3600 + 8=18 + 8 = 26\). So the minimum height is 8 feet and the maximum height (at the supports) is 26 feet.

Answer:

maximum height = 26 feet and minimum height = 8 feet (the last option: maximum height = 26 feet and minimum height = 8 feet)