Problem 81 - Contemporary Precalculus

Question

The cables of a suspension bridge are shaped like parabolas. The cables are attached to the towers 100 feet from the bridge surface, and the towers are 420 feet apart. The cables touch the bridge surface at the center (midway between the towers). At a point on the bridge 100 feet from one of the towers, how far is the cable from the bridge surface? (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

Answer: The approximate distance between the cable and the bridge surface at this point is 27.44 feet.

Step 1: Determine the given information

We're given that the towers' height is 100 feet from the bridge surface and they are 420 feet apart. The cables also touch the bridge surface midway between the towers, which means the vertex of the parabola will be at the point (420/2, 0) = (210, 0). The height of the parabola at the towers (x = 0 and x = 420) is 100.

Step 2: Find the equation of the parabola

To find the equation of the parabola, we can use the vertex form: y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Since the vertex is (210, 0), the equation will be y = a(x-210)^2. Now we need to find the value of 'a'. Since the towers' height is 100 feet, we have the points (0, 100) and (420, 100) on the parabola. Let's use the point (0, 100) to find 'a': 100 = a(0-210)^2 100 = a(-210)^2 100 = 44100a Divide both sides by 44100: a = 100/44100 a = 1/441 So the equation of the parabola is y = (1/441)(x-210)^2.

Step 3: Find the distance from the cable to the bridge surface

We need to find the distance between the cable and the bridge surface at a point 100 feet away from one of the towers, which means we need to calculate the y-coordinate when x = 100. y = (1/441)(100-210)^2 y = (1/441)(-110)^2 y = (1/441)(12100) y = 12100/441 y ≈ 27.44 Thus, the distance between the cable and the bridge surface at this point is approximately 27.44 feet.

Key Concepts

Equation of a Parabola

Parabolas are fascinating curves you often come across in the real world, especially in structures like suspension bridges. The equation of a parabola can take several forms, and knowing which one to use is crucial for solving different problems.
In the case of our suspension bridge, the cables follow a parabolic path. This path is described using a mathematical formula known as the equation of a parabola. One common form of this equation is the **vertex form**, which is helpful for identifying the key features of the parabola, such as its peak or low point.
In general, the vertex form of a parabola is represented as:

• \[ y = a(x - h)^2 + k \]

Here, each element in the equation plays a role:

• "\(a\)" determines the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• "\(h\)" and "\(k\)" represent the vertex of the parabola, where \((h, k)\) is its peak or the point of symmetry.

Knowing the coordinates of the vertex allows us to easily write the equation of the parabola, providing a clear map of how the parabolic cable hangs between the towers.

Vertex Form

The vertex form of a parabola is an especially useful format because it highlights the vertex. In mathematical terms, the vertex is where the parabola changes direction: its lowest point if it opens upwards, or its highest if it opens downwards.
The vertex form is given by:

• \[ y = a(x - h)^2 + k \]

The letters in this equation represent important values:

• "\(a\)" affects the parabola's width. Larger values make it narrower, while smaller values make it wider.
• "\(h\)" and "\(k\)" are the x and y coordinates of the vertex, establishing the symmetry of the parabola around the line \(x = h\).

In our suspension bridge cable problem, we know the vertex because the cable touches the bridge surface exactly halfway between the two towers. The towers are 420 feet apart, making the center 210 feet from either tower.
Substituting these into the formula gives an equation tailored to this specific situation. Ultimately, this helps us determine how the cable behaves at points away from the center, like the distance from the bridge surface at 100 feet.

Distance Calculation

Calculating distances can be a bit tricky, especially when dealing with parabolic curves. However, once we have the parabola's equation in vertex form, finding a specific distance becomes much simpler.
In the context of the suspension bridge, we're interested in how far the cable is from the bridge surface at specific points. To find this, we substitute the x-coordinate of the point of interest into the parabola's equation.
Using our established parabola:

• \[ y = \frac{1}{441}(x - 210)^2 \]

We're tasked with finding the distance when \(x = 100\) feet from one of the towers:

• Start by computing: \((100 - 210)^2\).
• Multiply the result by \(1/441\).
• Evaluate: \( y \approx 27.44 \).

This final result, 27.44, indicates that at 100 feet from the tower, the cable is about 27.44 feet above the bridge surface. Getting comfortable with these steps allows you to solve similar problems with ease, ensuring the parabolic equations don't just sit in textbooks but have practical applications too.